Keywords: lanthanum nickelate, oxygen electrode, Solid Oxide Electrolysis Cell, aging, energy efficiency

The present study focuses on an alternative structured oxygen electrode for solid oxide electrolysis cell (SOEC). La2NiO4+δ nickelate, LNO, was selected with respect to its mixed electronic and ionic conductivity and its good chemical stability. Complete single cells were built using screen printing of LNO and LaNi0.6Fe0.4O3 (LNF) layers on a standard hydrogen electrode supported half cell consisting of a porous NiO-YSZ layer, a dense 8YSZ electrolyte layer and a thin interlayer of GDC. The LNO cells polarization curves under various anodic flows of H2O/H2 (90/10) show excellent performances compared to the commercial LSCF (La0.6Sr0.4Co0.2Fe0.8O3-x) based cells. The steam conversion ratio SC increases from around 40% to 80% when the gas flow decreases from 36 to 12 NmL min -1 cm -2 . With a degradation rate of 1.5% 1,000 hrs -1 after 500 hrs operating time (i = -0.5 A cm -2 ) at 800 °C, using a medium H2O/H2 (90/10) gas flow of 24 NmL min -1 cm -2 , LNO oxygen electrode shows higher durability than the state of the art LSCF oxygen electrode. Energy efficiency analysis confirms the dependency of the total efficiency with the current density and gas flow allowing us to suggest optimized conditions for reaching targets of commercial H2 production.

Introduction

Nuclear energy systems usually operate at full capacity to produce electricity for grid base load power and the excess of heat and electricity should be recovered using an efficient way. For instance, the production of hydrogen (H2) as clean and renewable commodity for storing energy can be largely promoted through the high temperature steam electrolysis (HTSE) [1]. Indeed, for large-scale H2 production, high-temperature solid oxide electrolysis cells (SOECs) technology is considered as quite efficient [2][3][4]. However, to operate at high temperature, the SOEC components must meet some basic requirements for efficient and cost-effective hydrogen production, typically a good chemical stability in highly reducing/oxidizing environments and low reactivity versus. ageing. Yttria Stabilized Zirconia (YSZ) appears to be the most suitable electrolyte having a high ionic conductivity and being low cost. Ni-YSZ cermet as hydrogen electrode material and LSCF (La0.6Sr0.4Co0.2Fe0.8O3-x) as oxygen electrode material are most widely used and commercially available [5]. However, during long-term operation, a degradation of LSCF is observed due to Sr and Co depletion [6]. Hence, Sr and Co-free alternative anode materials, such as Ln2NiO4+δ (Ln = La, Nd, Pr) nickelates, have attracted the interest of researchers with regard to their good mixed ionic electronic conducting (MIEC) properties [7][8][9][10][11]. Indeed, these materials show high electronic and O 2-ionic conductivities and large oxygen surface exchange rates inducing excellent electrochemical properties for oxygen electrode applications [12][13][14][15][16][17]. An interesting feature of these compounds is the presence of additional oxygen in the layered structure leading to oxygen overstoichiometry (δ), which makes them chemically stable over a wide range of temperature and oxygen partial pressure pO2 [18][19][20][21]. Even though the praseodymium nickelate Pr2NiO4+δ exhibits the best performances, La2NiO4+δ (so-called LNO) appears the most stable material for use as an alternative oxygen electrode material for SOECs, especially with regard to aging [22][23]. This study aims to investigate and to highlight the performance of LNO as oxygen electrode in complete single cell in HTSE mode and to compare its behavior to the commercial anode material LSCF.

Experimental

Powder preparation and cell manufacturing

High purity powders of LNO were prepared via the citrate-nitrate route (modified Pechini method previously described in ref. [9]) starting from the precursors La2O3 (Strem Chemical, 99.99%), and Ni(NO3)2.6H2O (Acros Organics, 99%). The lanthanide oxide, La2O3, was prefired at 900 °C overnight to remove the water content because of its highly hygroscopic nature.

After the combustion step of the nitrate-citrate mixture, an annealing at 1200 °C for 12 hrs in air was carried out. Single and well crystallized phase was obtained as confirmed by X-ray diffraction.

Figure 1 shows a schematic view of the Ni-YSZ supported single cell. Planar hydrogen electrode supported half-cell (60 mm in diameter) was prepared by tape-casting for the support and by screen-printing for the H2 electrode functional layer and the electrolyte. The support was a 400 µm thick porous Ni-YSZ layer, the H2 functional layer was a 20 µm thick porous composite of nickel oxide and yttria-stabilized zirconia (NiO-YSZ) and the electrolyte was a 10 µm thick dense (98%) 8YSZ layer. NiO-YSZ and YSZ were sintered at 1400 °C during 5 hrs. The NiO/YSZ ratio was 66/34 wt.%. After reduction the porosity of the functional layer was close to 25%. The H2 electrode was reduced during 6 hrs before electrochemical measurements. A 3 µm thick barrier layer of Gadolinia Doped Ceria, Ce0.8Gd0.2O1.9, GDC was coated by screen-printing to limit the chemical reactivity between the electrolyte and the O2 electrode.

Terpineol-based LNO inks, containing 70% of active materials, were screen-printed on top of the GDC interface layer and sintered at 1200 °C, for 1 hr, in air, with a heating rate of 2 °C min - 1 , leading to layers of about 20 μm thick. The GDC interfacial layer and oxygen electrodes were respectively 55 and 20 mm in diameter, to use the edge of the cell for sealing. An additional layer of LaNi0.6Fe0.4O3 (LNF) was screen printed on LNO cell. This material was used as a collecting layer in order to improve the overall current collection. As previously shown [24], this material does not participate to the oxygen electrode reaction. It was in situ sintered in the electrochemical measurements setup at 900 °C for 2 hrs. These sintering temperatures were previously optimized to obtain a controlled homogeneous porous electrode microstructure corresponding to the lowest polarization resistance Rp values observed for LNO, (i.e., Rp < 0.10 Ω cm 2 at 800 °C) [25].

Electrochemical Measurements

Electrochemical characterizations were performed in a homemade set-up. In each experiment, cells were mounted and sealed to a metallic housing, providing gas flow channels and electrical connections. Anodic and cathodic gases were introduced at the cell center and distributed onto the electrodes. A glass seal was deposited on the edge of the cell to ensure a complete gas tightness of the hydrogen compartment. Platinum and nickel grids were used for the electrical contact (100 mesh cm -2 ), enhanced by a mechanical pressure of 400 g cm -2 applied on the setup. Steam was produced by burning the appropriate H2/O2 ratio in a specific chamber before feeding the cell. All the measurements were conducted using a Solartron potentiostat/impedance frequency analyzer (MODULAB with specific HV100 and FRA modules equipped with a booster 12 V 20 A). At the operating temperature and for the different gas feeding conditions, the polarization curves were recorded in galvanodynamic mode (20 mA s -1 ) from open circuit voltage (OCV) to 0.7 V in SOFC mode and starting from OCV to 1.4 V in SOEC mode. The cell aging was studied by recording the evolution of the voltage as a function of time in galvanostatic mode. Current density-voltage and power density curves were recorded at 800 °C in SOFC and SOEC modes. For SOFC mode the hydrogen flow (pure H2) was 24 NmL min -1 cm -2 while in SOEC mode the gas flow of H2O/H2 (90/10) mixture varies from 12 to 36 NmL min -1 cm -2 . The cell aging in SOEC mode was measured at 800 °C under a current density of -0.5 A cm -2 and a gas flow of 24 NmL min -1 cm -2 .

Results and Discussion

Start-up procedure

The test procedure started by heating the cell (under N2/H2 mixture for the H2 side and air for the O2 side) at a rate of 1 °C min -1 up to 920 °C during 30 minutes (glass sealing treatment).

After that, the temperature was lowered to the operating temperature, namely 800 °C, and the NiO/YSZ electrode was reduced by progressively increasing the hydrogen amount in the gas stream. The H2 electrode side was finally fed with only hydrogen gas using a flow of 24 NmL min -1 cm -2 . The corresponding oxygen partial pressure was considered close to p(O2) = 5×10 -24 bar. The OCV was measured in these conditions to evaluate the quality of the electrolyte tightness, and the correct formation of the glass seal before recording polarization curves in fuel cell mode (see Figure 2). Using the Nernst equation, a theoretical OCV value of 1.204 V was calculated. The closeness of the experimental OCV value (1.199 V) to the theoretical one validates the sample cell and the assembly in the testing set-up. In addition, the total Area Specific Resistance value of the cell (ASRtot) under operation determined in the linear domain (in the range 0.9 -0.7 V) was estimated at 0.285 Ω cm 2 . This relatively low value allowed concluding that the CERMET NiO/YSZ reduction process is satisfying and the cell is ready to be operated in the electrolysis mode. After these different steps, the cell was then fed by a H2O / H2 (90 /10) flow for experiments in electrolysis mode.

Electrochemical measurements in HTSE mode

At first, the OCV was measured in HTSE mode for various H2O/H2 flows (12 to 36 NmL min -1 cm -2 ). Compared to the theoretical value, 0.840 V, the determined experimental OCV values showed a relative difference, ∆OCV = (OCVexp.-OCVth.) /OCVth., lower than 1.7%, whatever the selected flow value (Table 1). Then, the polarization curves i-E in electrolysis mode were recorded for these three H2O/H2 flows. The data are plotted in Figure 3. The curves can be divided in three parts: in the low current density range (-0.20 < i < 0 A cm -2 ), the cells show a quasi nonlinear behavior. In the intermediate range, an ohmic-type behavior is observed in a large domain for the highest flow value, while for lower flows, they diverge from linearity at increasing current density absolute value, the linear part being quite limited when the flow is 12 NmL min -1 cm -2 . For an operating potential of 1.3 V corresponding to almost the auto thermal regime (∆H = 247.9 kJ mol -1 /2F), the current density is in the range -1.18 to -1.67 mA cm -2 as reported in Table 2. These values are rather close to the current density obtained with a similar cell using LSCF as anode material and operating in the same conditions (12-36 NmL min -1 cm -2 ). Concerning the highest current density, a limiting current density can be noticed.

Because this limit regime seems to appear for lower current when the supplying flow is lower it can be deduced that the mass transport (water vapor diffusion) is the limiting step and a steam depletion could be present at the electrode surface at the cathode side. These results allow to conclude on the good performance of the cell based on this anode architecture.

Considering a faradic efficiency equal to 1, the steam conversion ratio SC can be calculated from the operating current density divided by 2F according to the chemical reaction

H2O + 2e - H2 + O 2- (1)
and also the water vapor flow (taking into account that the gas mixture is 90% of water and 10% of hydrogen). As expected, the highest SC values are obtained for the lowest gas flow (around 82% for 12 NmL min -1 cm -2 ) and the SC drastically decreases when the gas flow increases (around 38% for 36 NmL min -1 cm -2 ).

The nonlinearity of the i-E curves observed in the current density range (-0.20 < i < 0 A cm - 2 ) (so called low regime), may be reasonably assigned to the electrolysis current that causes an increase in p(O2) at the air side, close to the electrode surface. It results in both an increase of the electronic and ionic conductivities directly correlated to an increase of the Ni 3+ amount (i.e., hole concentration) and additional oxide ions (i.e., increase in δ value). Such a behavior has been earlier reported by Kim et al. [26].

The values of the total ASR of the cell (ASRtot.) evaluated from the dE/di in the linear domain (intermediate regime) of the i-E curves (see Figure 3), are reported in Table 3. The ASRtot. value slightly decreases from 0.36 down to 0.25 Ω cm 2 when increasing the steam flow rate from 12 to 36 NmL min -1 cm -2 , respectively. These low values confirm the good performances of this cell in HTSE mode. A typical Electrochemical Impedance Spectroscopy (EIS) diagram measured under working atmospheres at 800 °C for the HTSE cells and at OCV is reported in Figure 4. Using such Nyquist plots, the intercepts between the impedance data and the real axis allow to deduce the series resistance R S(el+cc) and total ASRtot at high frequency and low frequency, respectively. Assuming that the total ASRtot. is the addition of the anode, electrolyte, cathode and current collectors impedance, one can analyze the relative contribution of each element at 800 °C. Indeed, the oxygen electrode polarization resistance RP(ox.) (LNO) was previously measured at 800 °C, in air, under zero dc conditions, and is close to 0.10 Ω cm 2 [27].

The ohmic resistance of the electrolyte and current collectors (RS(el + cc)) was measured around 0.13 Ω cm 2 using EIS. It can be mentioned here that the resistances values measured under zero dc conditions have been extrapolated under current. This hypothesis agrees with the high stability of the electrolyte and oxygen electrode impedances reported by previous authors [28].

Thus, by subtracting these contributions to the total resistance, ASRtot., the hydrogen electrode resistance RP(hy.) can be deduced. The data are reported in Table 3. The RP(hy.) contribution decreases with the steam flow supplying the electrode. Indeed, the more the active species (H2O) flow increases, the more the associated ASR at the hydrogen side decreases. For LNO, one should stress that for the lower gas flow rate, involving a high steam conversion, the hydrogen electrode contribution is much larger (around 36%). In such operating conditions, the oxygen electrode is no longer the most penalizing element of the cell (Rp(ox.) around 28%). A similar behavior is observed for LSCF based cell, confirming that the oxygen electrode is not always the most penalizing element of the cell, especially for high steam conversion ratios.

It is worthwhile comparing our results with those reported by Kim et al. [26] for similar cells, either with LSCF or LNO air electrodes, the current collector being a Pt mesh instead of LNF oxide. At 800 °C, the total ASR was 0.32 Ω cm 2 for LSCF and 0.43 Ω cm 2 for LNO, the flow rate of fuel consisting of 80% H2O/20% H2 of 100 NmL min -1 cm -2 was much higher than ours and the air electrode was exposed to open air. The LNO electrochemical cell tends to show better performances, probably due to an optimized cell manufacture. Both results confirm the promising properties of the lanthanum nickelate as efficient oxygen electrode material in SOEC.

Aging study under polarization

With regard to these good results of both cells in HTSE mode, the next step was to evaluate the aging issue. The selected operating conditions for that was a medium flow of 24 NmL min -1 cm -2 and a current density -0.5 A cm -2 (galvanostatic mode). Using these operating conditions, the steam conversion rate (SC) is close to 30%. The recorded voltage versus time (over 500 hours) is reported in Figure 5. The variation of voltage upon time shows a small decrease prior to an increase after about 50 hrs to finally reach a stabilization leading to a relative variation of around 1.5 % 1,000 hrs -1 after 500 hrs operating time. In comparison, the degradation is larger for cells with the LSCF oxygen electrode: during 250 hrs, the relative variation of voltage is around 2% 1,000 hrs -1 , then, it increases, being more than 4% 1,000 hrs -1 after 450 hrs. X-ray characterizations on both oxygen electrodes after aging show no evolution of the LNO electrode while extra SrCrO4 phase was detected in addition of LSCF. The long-term stability issues such as surface segregation of Sr during operation in LSCF is a well-documented phenomenon and a larger description is not the purpose of this paper. The absence of structural changes in LNO oxygen electrode after aging is in good agreement with its behavior in single cells as reported by Vibhu et al. [23]. For both oxygen electrodes, SEM analysis of the aged cells shows no major microstructure changes before and after aging as well as no clear evidence of adhesion loss in between the LNO electrode and the GDC electrolyte (See Figure 6). The purpose of this paper being the electrochemical investigation of LNO cells with comparison with LSCF cells, preliminary post-aging structural characterizations need to be deeper analyzed in the future.

Energy efficiency analysis

Even though the previous results suggest that the highest performances, in terms of current density-potential values, are obtained for a steam flow of 36 NmL min -1 cm -2 , it is relevant to evaluate the energy balance of these cells. The aim of this study is to determine the optimized operating conditions with respect to the steam flow and (i, E) parameters, starting from thermodynamic and polarization curves data.

The energy efficiency rate of a water electrolysis cell can be defined as the ratio of the energy produced by di-hydrogen gas combustion (low heating value, LHV) over the electrical energy amount injected to the cell to split the water molecules (current density i ≠ 0) under constant (T, p) operating conditions.

The total efficiency ρTot can be defined using the following equation [START_REF] O'hayre | Fuel Cell Fundamentals[END_REF]:

ρΤot. = ρThermo. × ρVoltage × ρ Faradic × SC (2)
where:

(i) ρThermo is the thermodynamic efficiency, which is defined as the ratio between the total enthalpy for water splitting (∆H) and the free enthalpy (∆G) at the operating temperature, i.e., ρThermo = ∆Hop. /∆Gop.. For T = 800 °C, their values are extracted from ref. [START_REF] Ruscic | Active Thermochemical Tables: water and water dimer[END_REF]: ∆Hop. = 247.9 kJ mol -1 and ∆Gop.= 188.1 kJ mol -1 . The value of ρThermo is then equal to 1.318. Such a high value (> 1), can be obtained because heat from the environment is transferred to the system during electrolysis operation, which can theoretically enable the production of more chemical energy of hydrogen fuel than the electrical energy supplied for the electrolysis process.

(ii) ρVoltage is the voltage efficiency. This term can be calculated using the ratio between the theoretical value of the open circuit voltage (OCV) (for given T and p(H2, H2O and O2)) and the real voltage of the cell under operating conditions E(i): ρVoltage = OCV / E(i). The reaction of water electrolysis being an endothermal process (∆Hop.> 0), thus, three operating modes can be distinguished as a function of the cell potential. At 800 °C, the thermodynamic potential of the reaction is close to 1.3 V: it corresponds to the so-called auto-thermal mode. For this specific potential value, the heat generated by the internal cell resistance is equal to the heat consumed by the electrolysis reaction. At a lower potential, heat must be externally supplied to maintain the system at a constant temperature, it is the endothermal mode. For cell potentials higher than 1.3 V, heat is generated by the internal cell resistance. (i.e., the Joule effect), the cell operates in the exothermal mode (it can be noted that this additional heat can be used to preheat the incoming gases at the system level).

For T = 800 °C and a cell fed in HTSE mode by a H2O/H2 (90/10) flow, the OCV is 0.840 V.

The calculation was performed for the operating conditions, Eop. = 1.3 V, and the corresponding current densities, iop., whose values are given in Table 4. It must be mentioned that these operating conditions have been considered in the intermediate regime, i.e., in the linear domain (ohmic domain) of the polarization curve (i, E). No mass transfer limitation or charge transfer activation regime has to be considered in these selected conditions.

(iii) ρFaradic is the ratio between the produced di-hydrogen amount versus the number of electrons injected in the cell. As currently observed with such HTSE cells, the ionic transport number of the ceramic membrane (YSZ) is close to one under the selected operating conditions (i.e., temperature, anodic and cathodic oxygen partial pressures, voltage and current density), then one can consider that ρFaradic = 1 [31].

(iv) SC represents the Steam Conversion as defined before. Concerning the fuel cell mode, the Fuel Utilization ratio (F.U. being the ratio between the theoretical H2 flow deduced from the electrical current and the experimental H2 injected flow) must be considered (see Table 4).

Using equation ( 2), the total efficiencies calculated for LNO and LSCF based SOECs are reported in Figure 7 versus the applied current density for different steam flows. As expected, three different regimes can be distinguished according to the values of the current density. For low negative current densities, so-called "low regime", from 0 to -0.5 A cm -2 , the total efficiency is limited by the low steam conversion ratio whatever the selected flow. At increasing negative current density, it is mainly governed by the voltage efficiency. For all the curves, in the high current density range, a maximum efficiency seems to be reached being a compromise between the contribution of these two parameters (SC and ρVoltage) (i.e., ρΤot = 69.3% at i = -1.28 A cm -2 , ρΤot. = 47.2% at i = -1.71 A cm -2 and ρΤot. = 35.7% at i ≈ -1.98 A cm -2 for 12, 24, 36 NmL min -1 cm -2 , respectively). Considering the only SOEC unit cell without taking into account auxiliary units, it can be concluded that an optimum efficiency is obtained using low steam supplying flow in association with a high (and adjusted) current density. Concerning the steam conversion ratio, the associated values are rather high: 81.5%, 51% and 38% for the 12, 24 and 36 NmL min -1 cm -2 flows, respectively. These raw SC values are in the range currently required for industrial application.

In addition, it can be noticed that the corresponding operating voltages, i.e. Eop = 1.395 V, 1.383 V and 1.391 V, for 12, 24 and 36 NmL min -1 cm -2 , respectively, are slightly higher than the thermo-neutral potential (1.285 V). It means that the cell would operate in the exothermal mode for which heat is generated by the internal cell resistance (i.e., Joule effect).

At this stage, the thermal operating conditions have to be discussed. Indeed, regarding the cell efficiency, a key point to consider is the energy associated to maintain the device at the operating temperature, i.e., 800 °C. In the previous calculation of ρTot.., the thermal energy necessary to heat and maintain the system at 800 °C was not taken into account. To consider this additional contribution, the thermal energy (T.∆S) should be introduced in the calculation using data of ref. [START_REF] Ruscic | Active Thermochemical Tables: water and water dimer[END_REF], i.e., T.∆S = -59.5 kJ mol -1 , i.e., 0.308 V, which leads to calculate the global efficiency # 2, ρGlob.. All the data and results are gathered in Table 4. The evolutions of ρTot. and ρGlob. versus the current density are reported for both cells in Figure 8. Such an analysis could be an interesting tool to help the operator to better identify the most appropriate operating conditions of the electrochemical cell (and according to the complete electrolysis system).

For both oxygen electrode materials, the efficiencies are very similar whatever the current density. As expected, when the thermal energy is considered the total efficiency is reduced up to ≈ 7% depending on the current density, i.e. the steam conversion ratio. According to the calculated efficiencies #1 and #2 reported in Table 4 at the thermo-neutral operating point (≈ 1.3

V), a strong influence of the supplying steam flow on the efficiency can be observed. In the investigated current density range (i.e., -2 < i (A cm -2 ) < 0), a low supplying flow allows improving drastically the energy efficiency.

In Table 4, the HTSE and SOFC modes are reported using a similar flow. First of all, from these data, it can be noticed that operating in the high temperature range, i.e., 800 °C, is very interesting for both HTSE and SOFC modes from a pure thermodynamic viewpoint because ρThermo stays high. Nevertheless, considering kinetic issues and under "classical" voltage operating conditions (i.e., 1.3 V and 0.7 V for HTSE and SOFC modes, respectively), the efficiency drastically decreases down to 43.5% for HTSE and 15.3% for SOFC mode (using the same supplying flow of 24 NmL min -1 cm -2 ). In the HTSE mode, even if the oxidationreduction reaction is theoretically endothermic in the low potential range, in practice, the electrolysis reaction will be exothermic in the operating condition because of various losses which have almost the same nature as in the fuel cell mode: activation and diffusion losses, and ohmic drop. In addition, when the thermal energy necessary for such high temperature conditions is taken into account, the total efficiency is again reduced. It is finally close to 56.1% for a water vapor flow of 12 NmL min -1 cm -2 .

According to studies dedicated to the complete HTSE systems (cf., Targets Europe [START_REF]Hydrogen Europe, Strategic Research and Innovation Agenda[END_REF]), a minimum steam conversion ratio close to 70% seems to be required for technical and economic reasons. Such a condition is obtained using the lowest water vapor flow (i.e., 12 NmL min -1 cm - 2 and i ≈ -1 A cm -2 ) whereas it is much more difficult to reach this target for a high flow range.

As an example, a current density close to -3 A cm -2 with 36 NmL min -1 cm -2 would be necessary to reach a SC of 70%.

Conclusion

Herein, the promising properties of the MIEC oxide, La2NiO4+δ, as oxygen electrode for SOECs are demonstrated through a careful analysis of its electrochemical properties. The polarization curves of LNF/LNO /GDC/8YSZ/ Ni-YSZ complete single cells, in HTSE mode, at 800 °C, show a strong dependence of the steam flow (H2O/H2 (90/10)), varying from 12 to 36 NmL min -1 cm -2 . In addition, from the determined ASRs, the contribution of each component of the cell is evaluated and it is clearly observed that the oxygen electrode is no more the most penalizing element at low flow rate. The comparison with LSCF based complete cells confirms the benefit of using LNO electrode: a degradation rate of 1.5 % 1,000 hrs -1 for 500 hrs aging is reported for LNO based complete cell under a medium flow of H2O/H2 (90/10) of 24 NmL min -1 cm -2 and a current density of -0.5 A cm -2 (galvanostatic mode) whereas it is close to 4 % 1,000 hrs -1 for LSCF based cell. The water vapor conversion rate (SC) also strongly depends on the steam flow rate. Then, a careful analysis of the energy efficiency led us to suggest operating conditions to reach the maximum SC close to 70% required for technical and economical reasons. Such a condition is obtained using a low water vapor flow (i.e., 12 NmL min -1 cm -2 and a current density of i ≈ -1.15 A cm -2 ).

Polarization curve and corresponding power density of the cell with La2NiO4+δ, LNO, oxygen electrode, in SOFC mode, at 800°C, under dry H2 flow (24 N ml min -1 cm -2 ) and air at the cathode side Aging performance of the cell, with either LNO oxygen electrode or LSCF oxygen electrode in hightemperature steam electrolysis (HTSE) mode, at 800°C, under H2O /H2 (90/10) flow of 24 N ml min -1 cm -2 and a current density of -0.5 A cm -2

FIGURE 6

Scanning electron microscopy (SEM) images of aged cells with LNO oxygen electrode. A magnification of the cell is given on the right

Dependence of the total efficiency on the current density for cells with LNO or LSCF oxygen electrodes in high-temperature steam electrolysis (HTSE) mode, at 800°C, under various H2O/H2 (90/10) flows

FIGURE 8

Current density dependence of the calculated efficiencies ρTot. and ρ Glob. for cells with LNO or LSCF oxygen electrode in high-temperature steam electrolysis (HTSE) mode, at 800°C, under an H2O/H2 (90/10) flow (36 N ml min -1 cm -2 )

Figure 1 -

 1 Figure 1 -Schematic description of the Ni-YSZ cermet supported single cell (F = 60 mm, CEA support) .

Figure. 2 -

 2 Figure. 2 -Polarization curve and corresponding power density of the cell with La2NiO4+δ, LNO, oxygen electrode, in SOFC mode, at 800 °C, under dry H2 flow (24 NmL min -1 cm -2 ) and air at the cathode side.

Figure. 3 -

 3 Figure. 3 -Polarization curves of cells with either LNO or LSFC as oxygen electrode in HTSE mode, at 800 °C, under various H2O/H2 (90/10) flows.

Figure 4 -

 4 Figure 4 -Typical EIS diagram of the electrochemical cells measured at 800 °C under zero dc condition.

Figure 5 -

 5 Figure 5 -Aging performance of the cell, with either LNO oxygen electrode or LSCF oxygen electrode in HTSE mode, at 800 °C, under H2O /H2 (90/10) flow of 24 NmL min -1cm -2 and a current density of -0.5 A cm -2 .

Figure 6 -Figure 7 -

 67 Figure 6 -SEM images of aged cell with LNO oxygen electrode. A magnification of the cell is given on the right

Figure 8 -

 8 Figure 8 -Current density dependence of the calculated efficiencies ρTot. and ρ Glob. for cells with LNO or LSCF oxygen electrode in HTSE mode, at 800 °C, under a H2O/H2 (90/10) flow (36 NmL min -1 cm -2 ).

FIGURE 1

 1 FIGURE 1Schematic description of the nickel-Yttria-stabilized zirconia (Ni-YSZ) cermet supported single cell (F = 60 mm, CEA support)

FIGURE 3 4

 34 FIGURE 3Polarization curves of cells with either LNO or LSCF as oxygen electrodes in high-temperature steam electrolysis (HTSE) mode, at 800°C, under various H2O/H2 (90/10) flows

FIGURE 5

 5 FIGURE 5

Table 1 :

 1 Experimental OCV values recorded using various H2O / H2 flows (12 ; 24 ; 36 NmL min -1 cm -2 ) and the relative difference ∆OCV/OCVth. (%).

	Flow / NmL min -1 cm -2	OCVexp. / V	(∆OCV) /OCVth. / %
	12	0.840	0
	24	0.832	-0.95
	36	0.826	-1.67

Table 2 :

 2 Current density i and steam conversion SC measured at 1.3 V

	Cells / Gas Flow	i at 1.3 V / A cm -2 SC at 1.3 V / %
	LNO -36 NmL min -1 cm -2	-1.67	38.5
	LNO -24 NmL min -1 cm -2	-1.48	51.1
	LNO -12 NmL min -1 cm -2	-1.18	81.5
	LSCF-36 NmL min -1 cm -2	-1.76	40.5
	LSCF-24 NmL min -1 cm -2	-1.56	53.9
	LSCF-12 NmL min -1 cm -2	-1.22	84.3

Table 3 :

 3 Total ASR of the cell, specific ASR of the different components and associated percentages

Gas Flow / NmL min -1 cm -2 ASRtot / Ω cm 2 RP(ox.) / Ω cm 2 Relative RP(ox.) % RS(el.+cc) / Ω cm 2 Relative RS(el.+cc) % RP(hy.) / Ω cm 2 Relative RP(hy.) %

	LNO-12	0.36	0.10	27.8	0.13	36.1	0.13	36.2
	LNO-24	0.29	0.10	33.9	0.13	44.0	0.06	22.0
	LNO-36	0.25	0.10	40.0	0.13	52.1	0.02	8.0
	LSFC-12	0.32	0.03	9.4	0.13	40.6	0.16	50.0
	LSFC-24	0.25	0.03	12.0	0.13	52.1	0.09	36.0
	LSFC-36	0.24	0.03	12.5	0.13	54.2	0.08	33.3

Table 4 :

 4 Energy efficiency of the cell under operating conditions (Eop., iop. and gas flow supplying the cell): thermodynamic efficiency, voltage efficiency and total efficiencies without (#1) and with (# 2) external heating of the cell at 800 °C.(*) F.U. : fuel utilization

	Oxygen electrode	LNO	LNO	LNO LSCF LSCF LSCF	LNO
	Mode	HTSE HTSE HTSE HTSE HTSE HTSE	SOFC
	Gas flow /NmL min -1 cm -2	12	24	36	12	24	36	24
	Eop. / V	1.3	1.3	1.3	1.3	1.3	1.3	0.7
	iop. / A cm -2	-1.179 -1.478 -1.673 -1.220 -1.562 -1.772	1.002
	SC %	81.5	51.1	38.5	84.3	54.0	40.5	F.U.=34.8*
	Thermodynamic efficiency %	131.8	131.8 131.8	131.8 131.8	131.8	75.9
	Voltage efficiency # 1 %	64.6	64.6	64.6	64.6	64.6	64.6	58.1
	Total efficiency # 1 , ρTot. %	69.4	43.5	32.8	71.8	45.9	34.5	15.3
	Voltage efficiency # 2 %	52.2	52.2	52.2	52.2	52.2	52.2	46.3
	Global efficiency # 2, ρGlob. %	56.1	35.2	26.5	58.0	37.1	27.9	12.2

Acknowledgments:

The authors wish to thank Alexandre Kucharski for technical support, and the French ANR for financial support _Grant Number ANR-13-PRGE-0008


J Lopez-Abad

L Nguyen
email: nguyenl@logique.jussieu.fr

THE OSCILLATION STABILITY PROBLEM FOR THE URYSOHN SPHERE: A COMBINATORIAL APPROACH

Keywords: June, 2007. 2000 Mathematics Subject Classification. Primary: 22F05. Secondary: 03E02, 05C55, 05D10, 22A05, 51F99 Topological groups actions, Oscillation stability, Ramsey theory, Metric geometry, Urysohn metric space

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for ℓ 2 in the context of the Urysohn space U. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.

Introduction.

The purpose of this note is to present several partial results related to the oscillation stability problem for the Urysohn sphere, a problem about to the geometry of the Urysohn space U which can, in some sense, be seen as an analog for U of the well-known distortion problem for ℓ 2 . This latter problem appeared after the following central result in geometric functional analysis established by Milman: For N ∈ ω strictly positive, let S N denote the unit sphere of the (N + 1)-dimensional Euclidean space and let S ∞ denote the unit sphere of the Hilbert space ℓ 2 . If X = (X, d X ) is a metric space, Y ⊂ X and ε > 0, let also (Y ) ε = {x ∈ X : ∃y ∈ Y d X (x, y) ε}.

Then:

Theorem (Milman [START_REF] Milman | A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies[END_REF]). Let γ be a finite partition of S ∞ . Then for every ε > 0 and every N ∈ ω, there is A ∈ γ and an isometric copy S N of S N in S ∞ such that S N ⊂ (A) ε . Whether Milman's theorem still holds when N is replaced by ∞ is the content of the distortion problem for ℓ 2 . Equivalently, if ε > 0 and f : S ∞ -→ R is bounded and uniformly continuous, is there a closed infinite-dimensional subspace V of ℓ 2 such that sup{|f (x) -f (y)| : x, y ∈ V ∩ S ∞ } < ε? This question remained unanswered for about 30 years, until the solution of Odell and Schlumprecht in [START_REF] Odell | The distortion problem[END_REF]: Theorem (Odell-Schlumprecht [START_REF] Odell | The distortion problem[END_REF]). There is a finite partition γ of S ∞ and ε > 0 such that no (A) ε for A ∈ γ includes an isometric copy of S ∞ . This result is traditionally stated in terms of oscillation stability, a concept coming from Banach space theory. However, it turns out that it can also be stated thanks to a new concept of oscillation stability due to Kechris, Pestov and Todorcevic introduced in [START_REF] Kechris | Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups[END_REF] and more fully developed in [START_REF] Pestov | Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena, Publicações matematicas[END_REF]. The global formulation of this notion takes place at a very general level and allows to capture various phenomena coming from combinatorics and functional analysis. Nevertheless, it can be presented quite simply in the realm of complete separable ultrahomogeneous metric spaces, where it coincides with the Ramsey-theoretic concept of approximate indivisibility. Recall that a metric space X is ultrahomogeneous when every isometry between finite metric subspaces of X can be extended to an isometry of X onto itself. Now, for ε 0, call a metric space X ε-indivisible when for every strictly positive k ∈ ω and every χ : X -→ k, there is i < k and X ⊂ X isometric to X such that X ⊂ ( ←χ {i}) ε .

Then X is approximately indivisible when X is ε-indivisible for every ε > 0, and X is indivisible when X is 0-indivisible. For example, in this terminology, the aforementioned theorem of Odell and Schlumprecht asserts that S ∞ is not approximately indivisible. However, in spite of this solution, it is sometimes felt that something essential is still to be discovered about the metric structure of S ∞ . Indeed, quite surprisingly, the proof leading to the solution is not based on an analysis of the intrinsic geometry of ℓ 2 . This fact is one of the motivations for the present note:

In this article, hoping that a better understanding of S ∞ might be hidden behind a general approach of approximate indivisibility, we study the approximate indivisibility problem for another complete, separable ultrahomogeneous metric space, the Urysohn sphere S, defined as follows: Up to isometry, it is the unique metric space to which every sphere of radius 1/2 in the Urysohn space U is isometric. Equivalently, it is, up to isometry, the unique complete separable ultrahomogeneous metric space with diameter 1 into which every separable metric space with diameter less or equal to 1 embeds isometrically. In this note, we try to answer the following question implicitly present in [START_REF] Kechris | Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups[END_REF] and explicitly stated in [START_REF] Hjorth | An oscillation theorem for groups of isometries[END_REF] and [START_REF] Pestov | Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena, Publicações matematicas[END_REF]:

Question. Is the Urysohn sphere S oscillation stable? That is, given a finite partition γ of S and ε > 0, is there A ∈ γ such that (A) ε includes an isometric copy of S?

Our approach here is combinatorial and follows the general intuition according to which the structure of complete separable ultrahomogeneous metric spaces can be approached via combinatorial means. This intuition is based on two ideas. The first one is that the combinatorial point of view is relevant for the study of countable ultrahomogeneous metric spaces in general. This idea is already central in the work of Fraïssé completed in the fifties, even though Fraïssé theory takes place at the level of relational structures and includes much more than metric spaces (for a reference on Fraïssé theory, see [START_REF] Fraïssé | Theory of relations[END_REF]). More recently, it was also rediscovered by Bogatyi in a purely metric context, see [START_REF] Bogatyi | Universal homogeneous rational ultrametric on the space of irrational numbers[END_REF] and [START_REF] Bogatyi | Metrically homogeneous spaces[END_REF]. The second idea is that the complete separable ultrahomogeneous metric spaces are closely linked to the countable ultrahomogeneous metric spaces. This connection also appears in Bogatyi's work but is on the other hand supported by the following result (which answers a question posed in [START_REF] Bogatyi | Metrically homogeneous spaces[END_REF]): Theorem 1. Every complete separable ultrahomogeneous metric space Y includes a countable ultrahomogeneous dense metric subspace.

For example, consider the rational Urysohn space U Q which can be defined up to isometry as the unique countable ultrahomogeneous metric space with rational distances for which every countable metric space with rational distances embeds isometrically. The Urysohn space U arises then as the completion of U Q , a fact which is actually essential as it is at the heart of several important contributions about U. In particular, in the original article [START_REF] Urysohn | Sur un espace métrique universel[END_REF] of Urysohn, the space U is precisely constructed as the completion of U Q which is in turn constructed by hand.

Similarly, the Urysohn sphere S arises as the completion of the so-called rational Urysohn sphere S Q , defined up to isometry as the unique countable ultrahomogeneous metric space with distances in Q ∩ [0, 1] into which every at most countable metric space with distances in Q ∩ [0, 1] embeds isometrically.

At first glance, such a representation is relevant with respect to the oscillation stability problem for complete separable ultrahomogeneous metric spaces because it provides a direct way to transfer an approximate indivisibility problem to an exact indivisibility problem. For example, in the present case, it naturally leads to the question (explicitly stated in [START_REF] Nešetřil | Metric spaces are Ramsey[END_REF] and in [START_REF] Pestov | Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena, Publicações matematicas[END_REF]) of knowing whether S Q is indivisible, a question which was answered recently by to Delhommé, Laflamme, Pouzet and Sauer in [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF], where a detailed analysis of metric indivisibility is provided and several obstructions to indivisibility are isolated. Cardinality is such an obstruction: A classical result in topology asserts that as soon as a metric space X is uncountable, there is a partition of X into two pieces such that none of the pieces includes a copy of the space via a continuous 1 -1 map. Unboundedness is another example: If a metric space X is indivisible, then its distance set is bounded. Now, it turns out that S Q avoids these obstacles but encounters a third one: For a metric space X, x ∈ X, and ε > 0, let λ ε (x) be the supremum of all reals l 1 such that there is an ε-chain (x i ) i n containing x and such that d X (x 0 , x n ) l. Then, define

λ(x) = inf{λ ε (x) : ε > 0}.
Theorem (Delhommé-Laflamme-Pouzet-Sauer [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF]). Let X be a countable metric space. Assume that there is x 0 ∈ X such that λ(x 0 ) > 0. Then X is not indivisible. Now, for S Q , it is easy to see that ultrahomogeneity together with the fact that the distance set contains 0 as an accumulation point imply that every point x in S Q is such that λ(x) = 1. It follows that:

Corollary (Delhommé-Laflamme-Pouzet-Sauer [3]). S Q is divisible.
This result put an end to the first attempt to solve the oscillation stability problem for S. Indeed, had S Q been indivisible, S would have been oscillation stable. But in the present case, the coloring which is used to divide S Q does not lead to any conclusion and the oscillation stability problem for S has to be attacked from another direction.

Here, following with the intuition that approximate indivisibility for S can be attacked via the study of the exact indivisibility of simpler spaces, we can show:

Theorem 2. S is 1/3-indivisible.
This result is obtained after having shown that the problem of approximate indivisibility for S can be reduced to a purely combinatorial problem involving a family (S m ) m 1 of countable metric spaces which in some sense approximate the space S. For m ∈ ω strictly positive, set

[0, 1] m := { k m : k ∈ {0, . . . , m}}.
Then S m is defined as follows: Up to isometry it is the unique countable ultrahomogeneous metric space with distances in [0, 1] m into which every countable metric space with distances in [0, 1] m embeds isometrically. Then:

Theorem 3.
The following are equivalent: (i) S is oscillation stable (equivalently, approximately indivisible).

(ii) For every strictly positive m ∈ ω, S m is 1/m-indivisible.

(iii) For every strictly positive m ∈ ω, S m is indivisible.

The paper is organized as follows: In section 2, we introduce the spaces S m and study their relationship with S. In particular, this leads us to a stronger version of Theorem 3. In section 3, we follow the different directions suggested by Theorem 3 and study the indivisibility as well as the 1/m-indivisibility properties of the spaces S m . We then show how these results can be used to derive Theorem 2. Finally, we close with a short section including some remarks about possible further studies while an Appendix provides a proof of Theorem 1.

Note: Item (iii) of Theorem 3 was recently proved by the N. W. Sauer and the second author. The Urysohn sphere is therefore oscillation stable.

Discretization.

The purpose of this section is to prove Theorem 3 and therefore to show that despite the unsuccessful attempt realized with S Q , the oscillation stability problem for S can indeed be understood via the study of the exact indivisibility of simpler spaces. The starting point of our construction consists in the observation that S Q is the first natural candidate because it is a very good countable approximation of S, but this good approximation is paradoxically responsible for the divisibility of S Q . In particular, the distance set of S Q is too rich and allows to create a dividing coloring. A natural attempt at that point is consequently to replace S Q by another space with a simpler distance set but still allowing to approximate S in a reasonable sense. In this perspective, general Fraïssé theory provides a whole family of candidates. Indeed, recall that for a strictly positive m ∈ ω, [0, 1] m denotes the set {k/m : k ∈ {0, . . . , m}}. Then one can prove that there is a countable ultrahomogeneous metric space S m with distances in [0, 1] m into which every countable metric space with distances in [0, 1] m embeds isometrically and that those properties actually characterize S m up to isometry. In other words, the spaces S m are really the analogs of S Q after having discretized the distance set Q ∩ [0, 1] with [0, 1] m . The intuition is then that in some sense, this should allow them to approximate S. This intuition turns out to be right, as shown by the following proposition whose proof is postponed to subsection 2.1: Proposition 1. For every strictly positive m ∈ ω, there is an isometric copy S m of S m inside S such that ( S m ) 1/m = S.

The spaces S m consequently appear as good candidates towards a discretization of the oscillation stability problem for S. However, it turns out that Proposition 1 is not of any help towards a proof of Theorem 3. For example, Proposition 1 does not imply alone that if for some strictly positive m ∈ ω, S m is indivisible, then S is 1/m-indivisible: Assume that χ : S -→ k. χ induces a coloring of S m so by indivisibility of S m there is S m ⊂ S m isometric to S m on which χ is constant. But how does that allow to obtain a copy of S? For example, are we sure that ( S m ) 1/m includes a copy of S? We are not able to answer this question, but recent results of J. Melleray in [START_REF] Melleray | On the geometry of Urysohn's universal metric space[END_REF] strongly suggest that ( S m ) 1/m really depends on the copy S m and can be extremely small. In particular, it may not include a copy of S. Thus, to our knowledge, Proposition 1 does not say anything about the oscillation stability of S, except maybe that the spaces S m 's are not totally irrelevant for our purposes.

Fortunately, the spaces S m do allow to go much further than Proposition 1 and are indeed relevant objects. In particular, they allow to reach the following equivalence, extending Theorem 3: Theorem 4. The following are equivalent:

(i) S is oscillation stable. (ii) S Q is approximately indivisible. (iii) For every strictly positive m ∈ ω, S m is 1/m-indivisible. (iv) For every strictly positive m ∈ ω, S m is indivisible.
Subsections 2.2 to 2.5 are devoted to the proof of this result. But before going deeper into the technical details, let us mention here that part of our hope towards the discretization strategy comes from the proof of a famous result in Banach space theory, namely Gowers' stabilization theorem for c 0 [START_REF] Gowers | Lipschitz functions on Classical spaces[END_REF], where combinatorial Ramsey-type theorems for the spaces FIN ± k and FIN k imply that the unit sphere S c0 of c 0 and its positive part S + c0 are approximately indivisible.

2.1. Proof of proposition 1. We start with a definition: Given a metric space

X = (X, d X ), a map f : X -→]0, +∞[ is Katȇtov over X when: ∀x, y ∈ X, |f (x) -f (y)| d X (x, y) f (x) + f (y).
Equivalently, one can extend the metric d X on X .

∪ {f } by defining, for every x, y in X, d X (x, f ) = f (x) and d X (x, y) = d X (x, y). The corresponding metric space is then written X . ∪ {f }. Here, the concept of Katȇtov map is relevant because of the following standard reformulation of the notion of ultrahomogeneity: Lemma 1. Let X be a countable metric space. Then X is ultrahomogeneous iff for every finite subspace F ⊂ X and every Katȇtov map

f over F, if F . ∪ {f } embeds into X, then there is y ∈ X such that for every x ∈ F, d X (x, y) = f (x).
This result will be used constantly throughout the proof. Now, some notation:

For m ∈ ω strictly positive, recall that [0, 1] m = {k/m : k ∈ {0, . . . , m}}. For α ∈ [0, 1], set also ⌈α⌉ m = min([α, 1] ∩ [0, 1] m ) = ⌈mα⌉ m ,
where ⌈x⌉ = min([x, ∞[∩Z) is the ceiling function. Since S is the metric completion of S Q , it is enough to show that for every strictly positive m ∈ ω, there is an isometric copy S m of S m inside S Q such that ( S m ) 1/m = S Q . This is achieved thanks to a back and forth argument. The following is the main idea.

Claim. Suppose that X ⊂ S Q is finite and embeddable in S m , and let y ∈ S Q X.

Then the mapping f = f X,y,m : X ∪ {y} →]0, ∞[ defined by f (x) = ⌈d S Q (x, y)⌉ m if x ∈ X and f (y) = max{⌈d S Q (x, y)⌉ m -d S Q (x, y) : x ∈ X} is Katȇtov.
Assume this claim is true. Fix (x n ) n∈ω an enumeration of S m and (y n ) n∈ω an enumeration of S Q . We are going to construct σ : ω → ω together with a set

S m = { x σ(n) : n ∈ ω} ⊂ S Q so that: (i) σ is a bijection. (ii) x σ(n) → x σ(n) defines an isometry.
(iii) For every n ∈ ω, {y i : i n} ⊂ ({ x i : i 2n + 1}) 1/m . Observe that, since σ is a permutation, (i) and (ii) guarantee that x n → x n defines a surjective isometry between S m and S m . On the other hand, (iii) guarantees that ( S m ) 1/m = S Q .

Let σ(0) = 0, x 0 = y 0 . Suppose now all data up to 2n already defined in the appropriate way, i.e. fulfilling the obvious partial versions of (i), (ii) and (iii). Let

σ(2n + 1) = min(ω {σ(i) : 0 i 2n}). Set also xσ(2n+1) ∈ S Q such that: ∀i ∈ {0, . . . , 2n}, d S Q (x σ(i) , xσ(2n+1) ) = d S m (x σ(i) , x σ(2n+1) ). Next, if y n ∈ { x σ(i) : i 2n + 1} 1/m ,
then we define σ(2n + 2) and x σ(2n+2) as we did for 2n + 1. Otherwise, let f be the Katȇtov map given by the previous claim when applied to X = {x σ(i) : 0 i 2n + 1} and y n . Let x ∈ S Q realizing f . Now observe that the map g defined on {x σ(i) :

0 i 2n + 1} by g(x σ(i) ) = f ( x σ(i) ) is Katȇtov with values in [0, 1] m , so σ(2n + 2) = min{k ∈ ω : ∀i ∈ {0, . . . , 2n + 1}, d S m (x σ(i) , x k ) = g(x σ(i) )}
is well defined and we set xσ(2n+2) = x.

We now turn to the proof of the claim. Fix x, x ′ ∈ X. We have to prove:

|f (x) -f (x ′ )| d S Q (x, x ′ ) f (x) + f (x ′ ) (1) |f (x) -f (y)| d SQ (x, y) f (x) + f (y) (2)
For (1): The right inequality is not a problem:

d S Q (x, x ′ ) d S Q (x, y) + d S Q (y, x ′ ) f (x) + f (x ′ ).
For the left inequality, we use the following simple fact:

∀α, β ∈ R, ∀p ∈ ω, |β -α| p m -→ |⌈β⌉ m -⌈α⌉ m | p m .
Indeed, assume that |β -α| p/m. We want |⌈mβ⌉ -⌈mα⌉| p. Without loss of generality, α β. Then 0 ⌈mβ⌉ -⌈mα⌉ < mβ + 1 -mα p + 1, so |⌈mβ⌉ -⌈mα⌉| p and we are done. In our case, that property is useful because then the left inequality directly follows from

d SQ (x, y) -d SQ (y, x ′ ) d SQ (x, x ′ ) ∈ [0, 1] m , because X is embeddable in S m . For (2): |f (x) -f (y)| = f (x) -f (y).
This is because f (x) 1/m and 0 f (y) < 1/m. Furthermore, by definition of f ,

f (y) f (x) -d S Q (x, y).
So the left inequality is satisfied. For the right inequality, simply observe that d SQ (x, y) f (x).

2.2.

From oscillation stability of S to approximate indivisibility of S Q .

The purpose of what follows is to prove the implication (i) → (ii) of Theorem 4 stating that if S is oscillation stable, then S Q is approximately indivisible. This is done thanks to the following result:

Proposition 2. Suppose that S 0 Q and S 1 Q are two copies of S Q in S such that S 0 Q is dense in S. Then for every ε > 0 the subspace S 0 Q ∩ (S 1 Q ) ε includes a copy of S Q . Proof. We construct the required copy of S Q inductively. Let {y n : n ∈ ω} enumer- ate S 1 Q . For k ∈ ω, set δ k = ε 2 k i=0 1 2 i .
Set also

η k = ε 3 1 2 k+1 . S 0 Q being dense in S, choose z 0 ∈ S 0 Q such that d S (y 0 , z 0 ) < δ 0 . Assume now that z 0 , . . . , z n ∈ S 0 Q were constructed such that for every k, l n d S (z k , z l ) = d S (y k , y l ) d S (z k , y k ) < δ k .
Again by denseness of

S 0 Q in S, fix z ∈ S 0 Q such that d S (z, y n+1 ) < η n+1 .
Then for every k n,

d S (z, z k ) -d S (y n+1 , y k ) = d S (z, z k ) -d S (z k , y n+1 ) + d S (z k , y n+1 ) -d S (y n+1 , y k ) d S (z, y n+1 ) + d S (z k , y k ) < η n+1 + δ k < η n+1 + δ n .
It follows that there is

z n+1 ∈ S 0 Q such that ∀k n d S (z n+1 , z k ) = d S (y n+1 , y k ) d S (z n+1 , z) < η n+1 + δ n .
Indeed, consider the map f defined on {z k : k n} ∪ {z} by:

∀k n f (z k ) = d S (y n+1 , y k ) f (z) = d S (z, z k ) -d S (y n+1 , y k ) . Then f is Katȇtov over the subspace of S 0 Q supported by {z k : k n} ∪ {z}, so simply take z n+1 ∈ S 0 Q realizing it. Observe then that d S (z n+1 , y n+1 ) d S (z n+1 , z) + d S (z, y n+1 ) < η n+1 + δ n + η n+1 < δ n+1 .
After ω steps, we are left with

{z n : n ∈ ω} ⊂ S 0 Q ∩ (S 1 Q ) ε isometric to S Q .
We now show how to deduce (i) → (ii) of Theorem 4 from Proposition 2: Let ε > 0, k ∈ ω strictly positive and χ : S Q -→ k. Then in S, seeing S Q as a dense subspace:

S = i<k ( ← - χ {i}) ε/2 .
By oscillation stability of S, there is i < k and a copy S of S included in S such that S ⊂ ((

← - χ {i}) ε/2 ) ε/4 .
Since S includes copies of S Q , and since S Q is dense in S, it follows by Proposition 2 that there is a copy

S Q of S Q in S Q ∩ ( S) ε/4 . Then in S Q S Q ⊂ ( ← - χ {i}) ε . 2.3. From approximate indivisibility of S Q to 1/m-indivisibility of S m .
Here, we provide a proof for the implication (ii) → (iii) of Theorem 4 according to which if S Q is approximately indivisible, then S m is 1/m-indivisible for every strictly positive m ∈ ω. This is obtained as the consequence of the following proposition:

Proposition 3. Let ε > 0 and assume that S Q is ε-indivisible. Then S m is 1/m- indivisible whenever m 1/ε. Proof. Let ε > 0, assume that S Q is ε-indivisible and fix m ∈ ω strictly positive such that ε 1/m. Define d S Q m by ∀x, y ∈ X d S Q m (x, y) = d S Q (x, y) m . Claim. d S Q
m is a metric on S Q . Proof. Since the function ⌈•⌉ m is subadditive and increasing, it easily follows that the composition ⌈d

S Q ⌉ m = ⌈•⌉ m • d S Q is a metric.
Let X m be the metric space

X m = (S Q , d SQ m )
, and let π m denote the identity map from S Q to X m . Observe that X m and S m embed into each other, and that consequently, 1/m-indivisibility of S m is equivalent to 1/m-indivisibility of X m . So let k ∈ ω be strictly positive and χ :

X m -→ k. Then χ induces a coloring χ • π m : S Q -→ k. Since S Q is ε-indivisible, there is i < k and a copy S Q of S Q inside S Q such that S Q ⊂ ( ← ---- χ • π m {i}) ε . Now, observe that π ′′ m S Q is a copy of X m inside X m . Furthermore, note that ∀x = y ∈ S Q if d S Q (x, y) 1 m then d X m (π m (x), π m (y)) = 1 m .
Since ε 1/m, it follows that

π ′′ m ( ← ---- χ • π m {i}) ε ⊂ ( ← - χ {i}) 1/m .
And so Assuming that such a space W is constructed, the result is proved as follows: Observe first that W and S m embed into each other. Indivisibility of W is consequently equivalent to indivisibility of S m and it is enough to show that W is indivisible. Let k ∈ ω be strictly positive and χ :

π ′′ m S Q ⊂ ( ← - χ {i})
W -→ k. Then χ•π : S 2(m 2 +m) -→ k and by 1/2(m 2 + m)-indivisibility of S 2(m 2 +m) , there is i < k such that ( ← --- χ • π{i}) 1/2(m 2 +m) includes a copy of S 2(m 2 +m) . Since S m embeds into S 2(m 2 +m) , ( ← --- χ • π{i}) 1/2(m 2
+m) also includes a copy of S m . Thus, ←χ {i} = π ′′← --χ • π{i} includes a copy of S m , and therefore a copy of W.

We now turn to the construction of W. This space is obtained by modifying the metric on S 2(m 2 +m) to a metric d, so that W = (S 2(m 2 +m) , d) and π is simply the identity map from S 2(m 2 +m) to W. The metric d is defined as follows: consider the map f :

[0, 1] 2(m 2 +m) -→ [0, 1] m defined by f (x) = l
m where l is the least integer such that

x l 1 m + 1 m 2 + m .
Observe that f is increasing, that f (0) = 0, and that

∀α ∈ [0, 1] m ∀ε ∈ {-2, -1, 0, 1, 2} f α + ε 2(m 2 + m) = α.
Note also that f is subadditive: Let x, y, ∈ [0, 1] 2(m 2 +m) . Assume that f (x) = l/m. Then there is n ∈ {1, . . . , 2m + 4} such that

x = l -1 m + l -1 m 2 + m + n 2(m 2 + m) .
Similarly, there are l ′ ∈ {0, . . . , m} and n ′ ∈ {1, . . . , 2m + 4} such that

y = l ′ -1 m + l ′ -1 m 2 + m + n ′ 2(m 2 + m) . So x + y = (l + l ′ ) 1 m + 1 m 2 + m -2 1 m + 1 m 2 + m + n + n ′ 2(m 2 + m) = (l + l ′ ) 1 m + 1 m 2 + m + n -(2m + 4) + n ′ -(2m + 4) 2(m 2 + m) (l + l ′ ) 1 m + 1 m 2 + m .
Therefore,

f (x + y) l + l ′ m = l m + l ′ m = f (x) + f (y).
It follows that the map d := f • d S 2(m 2 +m) is a metric taking values in [0, 1] m . Now to show that d is as required, it suffices to prove that for every subspace Y of S 2(m 2 +m) , if (Y) 1/2(m 2 +m) includes a copy of S m , then π ′′ Y includes a copy of S m . So let Y be a subspace of S 2(m 2 +m) such that (Y) 1/2(m 2 +m) includes a copy S m of S m . Then for every x ∈ S m , there is an element ϕ(x) ∈ Y such that d S 2(m 2 +m) (x, ϕ(x)) 1/2(m 2 + m). Thus,

∀x = y ∈ S m d S 2(m 2 +m) (ϕ(x), ϕ(y)) -d S 2(m 2 +m) (x, y) 1 m 2 + m . Since d S 2(m 2 +m) (x, y) ∈ [0, 1] m , f d S 2(m 2 +m) (ϕ(x), ϕ(y)) = d S 2(m 2 +m) (x, y).
That is

d(π(ϕ(x)), π(ϕ(y))) = d S 2(m 2 +m) (x, y).
Thus, π ′′ ran(ϕ) ⊂ π ′′ Y is isometric to S m .

2.5.

From indivisibility of S m to oscillation stability of S. We are now ready to close the loop of implications of Theorem 4. In what follows, we show that if S m is indivisible for every strictly positive m ∈ ω, then S is oscillation stable. This is achieved thanks to the following result:

Proposition 5. Assume that for some strictly positive m ∈ ω, S m is indivisible. Then S is 1/m-indivisible.
Proof. This is obtained by showing that for every strictly positive m ∈ ω, there is an isometric copy S * m of S m inside S such that for every S m ⊂ S * m isometric to S m , ( S m ) 1/m includes an isometric copy of S. This property indeed suffices to prove Proposition 5: Let χ : S -→ k for some strictly positive k ∈ ω. χ induces a k-coloring of the copy S * m . By indivisibility of S m , find i < k and S m ⊂ S * m such that χ is constant on S m with value i. But then, in S, ( S m ) 1/m includes a copy of S. So ( ←χ {i}) 1/m includes a copy of S. We now turn to the construction of S * m . The core of the proof is contained in Lemma 2 which we present now. Fix an enumeration {y n : n ∈ ω} of S Q . Also, keeping the notation introduced in the proof of Proposition 3, let X m be the metric space (S Q , d S Q m ). The underlying set of X m is really {y n : n ∈ ω} but to avoid confusion, we write it {x n : n ∈ ω}, being understood that for every n ∈ ω, x n = y n . On the other hand, remember that S m and X m embed isometrically into each other.

Lemma 2. There is a countable metric space Z with distances in [0, 1] and including X m such that for every strictly increasing σ : ω -→ ω such that

x n → x σ(n) is an isometry, ({x σ(n) : n ∈ ω}) 1/m includes an isometric copy of S Q .
Assuming Lemma 2, we now show how we can construct S * m . Z is countable with distances in [0, 1] so we may assume that it is a subspace of S. Now, take S * m a subspace of X m and isometric to S m . We claim that S * m works: Let S m ⊂ S * m be isometric to S m . We first show that ( S m ) 1/m includes a copy of S Q . The enumeration {x n : n ∈ ω} induces a linear ordering < of S m in type ω. According to lemma 2, it suffices to show that ( S m , <) includes a copy of {x n : n ∈ ω} < seen as an ordered metric space. To do that, observe that since X m embeds isometrically into S m , there is a linear ordering < * of S m in type ω such that {x n : n ∈ ω} < embeds into (S m , < * ) as ordered metric space. Therefore, it is enough to show:

Claim. ( S m , <) includes a copy of (S m , < * ).

Proof. Write (S m , < * ) = {s n : n ∈ ω} < * ( S m , <) = {t n : n ∈ ω} < . Let σ(0) = 0. If σ(0) < • • • < σ(n) are chosen such that s k → t σ(k) is a finite isometry, observe that the following set is infinite {i ∈ ω : ∀k n d Sm (t σ(k) , t i ) = d Sm (s k , s n+1 )}.
Therefore, simply take σ(n + 1) in that set and larger than σ(n).

Observe that since the metric completion of S Q is S, the closure of ( S m ) 1/m in S includes a copy of S. Hence we are done since ( S m ) 1/m is closed in S.

We now turn to the proof of lemma 2. The strategy is first to provide the set Z where the required metric space Z is supposed to be based on, and then to argue that the distance d Z can be obtained (lemmas 3 to 7). To construct Z, proceed as follows: For t ⊂ ω, write t as the strictly increasing enumeration of its elements: t = {t i : i ∈ |t|} < . Now, let T be the set of all finite nonempty subsets t of ω such that x n → x tn is an isometry between {x n : n ∈ |t|} and {x tn : n ∈ |t|}. This set T is a tree when ordered by end-extension. Let

Z = X m . ∪ T . For z ∈ Z, define π(z) = z if z ∈ X m .
x max z if z ∈ T . Now, consider an edge-labelled graph structure on Z by defining δ with domain dom(δ) ⊂ Z × Z and range ran(δ) ⊂ [0, 1] as follows:

• If s, t ∈ T , then (s, t) ∈ dom(δ) iff s and t are < T comparable. In this case, δ(s, t) = d S Q (y |s|-1 , y |t|-1 ). • If x, y ∈ X m , then (x, y) is always in dom(δ) and δ(x, y) = d Xm (x, y).

• If t ∈ T and x ∈ X m , then (x, s) and (s, x) are in dom(δ) iff x = π(t). In this case δ(x, s) = δ(s, x) = 1 m .
For a branch b of T and i ∈ ω, let b(i) be the unique element of b with height i in T . Observe that b(i) is a i + 1-element subset of ω. Observe also that for every i, j ∈ ω, b(i) is connected to π(b(i)) and b(j), and

(i) δ(b(i), π(b(i)) = 1/m, (ii) δ(b(i), b(j)) = d S Q (y i , y j ), (iii) δ(π(b(i)), π(b(j))
) is equal to any of the following quantities:

d X m (x max b(i) , x max b(j) ) = d X m (x i , x j ) = ⌈d S Q (y i , y j )⌉ m .
In particular, if b is a branch of T , then δ induces a metric on b and the map from S Q to b mapping y i to b(i) is a surjective isometry. We claim that if we can show that δ can be extended to a metric d Z on Z with distances in [0, 1], then lemma 2 will be proved. Indeed, let

X m = {x σ(n) : n ∈ ω} ⊂ X m ,
with σ : ω -→ ω strictly increasing and x n → x σ(n) distance preserving. See ran(σ) as a branch b of T . Then (b,

d Z ) = (b, δ) is isometric to S Q and b ⊂ (π ′′ b) 1/m = ( X m ) 1/m .
Our goal now is consequently to show that δ can be extended to a metric on Z with values in [0, 1]. Recall that for x, y ∈ Z, and n ∈ ω strictly positive, a path from x to y of size n as is a finite sequence γ = (z i ) i<n such that z 0 = x, z n-1 = y and for every i < n -1, (z i , z i+1 ) ∈ dom(δ). For x, y in Z, P (x, y) is the set of all paths from x to y. If γ = (z i ) i<n is in P (x, y), γ is defined as:

γ = n-1 i=0 δ(z i , z i+1 ).
On the other hand, γ 1 is defined as:

γ 1 = min( γ , 1).
We are going to see that the required metric can be obtained with d Z defined by

d Z (x, y) = inf{ γ 1 : γ ∈ P (x, y)}.
Equivalently, we are going to show that for every (x, y) ∈ dom(δ), every path γ from x to y is metric, that is:

δ(x, y) γ 1 (3)
Let x, y ∈ Z. Call a path γ from x to y trivial when γ = (x, y) and irreducible when no proper subsequence of γ is a non-trivial path from x to y. Finally, say that γ is a cycle when (x, y) ∈ dom(δ). It should be clear that to prove that d Z works, it is enough to show that the previous inequality ( 3) is true for every irreducible cycle. Note that even though δ takes only rational values, it might not be the case for d Z .

We now turn to the study of the irreducible cycles in Z.

Lemma 3. Let x, y ∈ T . Assume that x and y are not < T -comparable. Let γ be an irreducible path from x to y in T . Then there is z ∈ T such that z < T x, z < T y and γ = (x, z, y).

Proof. Write γ = (z i ) i<n+1 . z 1 is connected to x so z 1 is < T -comparable with
x. We claim that z 1 < T x : Otherwise, x < T z 1 and every element of T which is < T -comparable with z 1 is also < T -comparable with x. In particular, z 2 is < Tcomparable with x, a contradiction since z 2 and x are not connected. We now claim that z 1 < T y. Indeed, observe that z 1 < T z 2 : Otherwise, z 2 < T z 1 < T x so z 2 < T x contradicting irreducibility. Now, every element of T which is < T -comparable with z 2 is also < T -comparable with z 1 , so no further element can be added to the path. Hence z 2 = y and we can take z 1 = z. Proof. Let c = (z i ) i<n be a non-trivial irreducible cycle in T . We may assume that z 0 < T z n-1 . Now, observe that every element of T comparable with z 0 is also comparable with z n-1 . In particular, z 1 is such an element. It follows that n = 3 and that z 0 , z 1 , z 2 are in a same branch.

Lemma 6. Every irreducible cycle in Z intersecting both X m and T is supported by a set whose form is one of the following ones. If C ∩ X m has size 1, let z 0 be its unique element. In c, z 0 is connected to two elements which we denote z 1 and z 3 . Note that z 1 , z 3 ∈ T so π(z 1 ) = π(z 3 ) = z 0 . Since elements in T which are connected never project on a same point, it follows that z 1 , z 3 are < T -incomparable. Now, c induces an irreducible path from z 1 to z 3 in T so from lemma 3, there is z 2 ∈ C such that z 2 < T z 1 , z 2 < T z 3 , and we are in case 2.

Assume now that C ∩ X m = {z 0 , z 4 }. Then there are z 1 , z 3 ∈ C ∩ T such that π(z 1 ) = z 0 and π(z 3 ) = z 4 . Note that since z 0 = z 4 , we must have z 1 = z 3 . Now, C ∩ T induces an irreducible path from z 1 to z 3 in T . By lemma 3, either z 1 and z 3 are compatible and in this case, we are in case 1, or z 1 and z 3 are < T -incomparable and there is z 2 in C ∩ T such that z 2 < T z 1 , z 2 < T z 3 and we are in case 3.

Lemma 7. Every non-trivial irreducible cycle in Z is metric.

Proof. Let c be an irreducible cycle in Z. If c is supported by X m , then by lemma 4 c has size 3 and is metric since δ induces a metric on X m . If c is supported by T , then by lemma 5 c also has size 3 and is included in a branch b of T . Since δ induces a metric on b, c is metric. We consequently assume that c intersects both X m and T . According to lemma 6, c is supported by a set C whose form is covered by one of the cases 1, 2 or 3. So to prove the present lemma, it is enough to show every cycle obtained from a re-indexing of the cycles described in those cases is metric.

Case 1: The required inequalities are obvious after having observed that

δ(z 0 , z 3 ) = ⌈δ(z 1 , z 2 )⌉ m and δ(z 0 , z 1 ) = δ(z 2 , z 3 ) = 1 m .
Case 2: Notice that δ(z 0 , z 1 ) = δ(z 0 , z 3 ) = 1/m. So the inequalities we need to prove are

δ(z 1 , z 2 ) δ(z 2 , z 3 ) + 2 m , (4)
δ(z 2 , z 3 ) δ(z 1 , z 2 ) + 2 m . (5)
By symmetry, it suffices to verify that (4) holds. Observe that since π(z

1 ) = π(z 3 ) = z 0 , we must have ⌈δ(z 1 , z 2 )⌉ m = ⌈δ(z 2 , z 3 )⌉ m . So: δ(z 1 , z 2 ) ⌈δ(z 1 , z 2 )⌉ m = ⌈δ(z 2 , z 3 )⌉ m δ(z 2 , z 3 ) + 2 m .
Case 3: Observe that δ(z 0 , z 1 ) = δ(z 3 , z 4 ) = 1/m, so the inequalities we need to prove are

δ(z 1 , z 2 ) δ(z 2 , z 3 ) + δ(z 0 , z 4 ) + 2 m , (6)
δ(z 0 , z 4 ) δ(z 1 , z 2 ) + δ(z 2 , z 3 ) + 2 m . (7) For (6)
:

δ(z 1 , z 2 ) ⌈δ(z 1 , z 2 )⌉ m = δ(π(z 1 ), π(z 2 )) = δ(z 0 , π(z 2 )) δ(z 0 , z 4 ) + δ(z 4 , π(z 2 )) = δ(z 0 , z 4 ) + ⌈δ(z 3 , z 2 )⌉ m δ(z 0 , z 4 ) + δ(z 2 , z 3 ) + 2 m . For (7): Write z 1 = b(j), z 3 = b ′ (k), z 2 = b(i) = b ′ (i). Then z 0 = π(z 1 ) = x max b(j) and z 4 = π(z 3 ) = x max b ′ (k) .
Observe also that δ(z 1 , z 2 ) = d SQ (y j , y i ) and that δ(z 2 , z 3 ) = d SQ (y i , y k ). So:

δ(z 0 , z 4 ) = d Xm (x max b(j) , x max b ′ (k) ) d X m (x max b(j) , x max b(i) ) + d X m (x max b ′ (i) , x max b ′ (k) ) = d Xm (x j , x i ) + d Xm (x i , x k ) = d S Q (y j , y i ) m + d S Q (y i , y k ) m = ⌈δ(z 1 , z 2 )⌉ m + ⌈δ(z 2 , z 3 )⌉ m δ(z 1 , z 2 ) + 1 m + δ(z 2 , z 3 ) + 1 m = δ(z 1 , z 2 ) + δ(z 2 , z 3 ) + 2 m .

Results and bounds.

Ideally, the title of this section would have been "The Urysohn sphere is oscillation stable" and we would have ended this article with the proof of one of the different formulations of oscillation stability for S presented in Theorem 4. Unfortunately, so far, our numerous attempts to reach this goal did not succeed 1 . This is why this part is entitled "bounds". Instead, what we will be presenting now will show how far we were able to push in the different directions suggested by Theorem 4. We start with a summary about the indivisibility properties of the spaces S m .

3.1. Are the S m 's indivisible? Of course, when m = 1, the space S m is indivisible in virtue of the most elementary pigeonhole principle on ω. The first non-trivial case is consequently for m = 2. However, this case is also easy to solve after having noticed that S 2 is really the Rado graph R where the distance is 1/2 between connected points and 1 between non-connected distinct points. Therefore, indivisibility for S 2 is equivalent to indivisibility of R, a problem whose solution is well-known: Proposition 6. The Rado graph R is indivisible.

The following case to consider is S 3 , which turns out to be another particular case thanks to an observation made in [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF]. Indeed, S 3 can be encoded by the countable ultrahomogeneous edge-labelled graph with edges in {1/3, 1} and forbidding the complete triangle with labels 1/3, 1/3, 1. The distance between two points connected by an edge is the label of the edge while the distance between two points which are not connected is 2/3. This fact allows to show:

Theorem (Delhommé-Laflamme-Pouzet-Sauer [3]). S 3 is indivisible.
Indeed, the proof of this theorem can be deduced from the proof of the indivisibility of the K n -free ultrahomogeneous graph by El-Zahar and Sauer in [START_REF] El-Zahar | The indivisibility of the homogeneous Kn-free graphs[END_REF]. We do not write more here but the interested reader is referred to [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF], section on the indivisibility of Urysohn spaces, for more details.

The very first substantial case consequently shows up for m = 4. Unfortunately, it appears to be so substantial that so far, we still do not know whether this space is indivisible or not. Nevertheless, we are able to establish that if this space is not indivisible, then S 4 is quite exceptional, in a sense that we precise now. We already mentioned that [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF] contains an analysis of indivisibility in the realm of countable metric spaces. It turns out that this study also led its authors to examine the conditions under which a set of strictly positive reals can be interpreted as the distance set of a countable universal and ultrahomogeneous metric space: Definition (4-values condition). Let S ⊂]0, +∞[. S satisfies the 4-values condition when for every s 0 , s 1 , s ′ 0 , s ′ 1 ∈ S, if there is t ∈ S such that:

|s 0 -s 1 | t s 0 + s 1 and |s ′ 0 -s ′ 1 | t s ′ 0 + s ′ 1 , then there is u ∈ S such that: |s 0 -s ′ 0 | u s 0 + s ′ 0 and |s 1 -s ′ 1 | u s 1 + s ′ 1 .
Theorem (Delhommé-Laflamme-Pouzet-Sauer [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF]). Let S ⊂]0, +∞[. TFAE:

1 The goal has now been achieved by N. W. Sauer and the second author (i) There is a countable ultrahomogeneous metric space U S with distances in S into which every countable metric space with distances in S embeds isometrically. (ii) S satisfies the 4-values condition.

As detailed in [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF], the 4-values condition covers a wide variety of examples. For our purposes, the 4-values condition is relevant because it allows to establish a list of spaces such that any space U S with S finite is in some sense isomorphic to some space in the list. In particular, it allows to set up a finite list of spaces exhausting all the spaces U S with S 4. More precisely, for finite subsets S = {s 0 , . . . , s m } < , T = {t 0 , . . . , t n } < of ]0, +∞[, define S ∼ T when m = n and: ∀i, j, k < m, s i s j + s k ↔ t i t j + t k .

Observe that when S ∼ T , S satisfies the 4-value condition iff T does and in this case, S and T essentially provide the same metric spaces as it is possible to have U S and U T supported by ω with the metrics d U S and d U T being defined such that: ∀x, y ∈ ω, d US (x, y) = s i ↔ d UT (x, y) = t i . Now, clearly, for a given cardinality m there are only finitely many ∼-classes, so we can find a finite collection S m of finite subsets of ]0, ∞[ of size m such that for every T of size m satisfying the 4-value condition, there is S ∈ S m such that T ∼ S. For m 3, examples of such lists can be easily provided. For instance, one may take:

S 1 = {{1}} S 2 = {{1, 2}, {1, 3}} S 3 = {{2, 3, 4}, {1, 2, 3}, {1, 2, 5}, {1, 3, 4}, {1, 3, 6}, {1, 3, 7}}.
Notice that in those lists, the set [0, 1] m is represented by the set {1, 2, . . . , m}. For m = 4, a long and tedious checking procedure of the 4-values condition allows to find S m but it then contains more than 20 elements so there is no point writing them all here. Still, it turns out that in most of the cases, we are able to solve the indivisibility problem for the space U S . Our result can be stated as follows: Due to the number of cases to consider, we do not prove this theorem here but simply mention that when the proofs are not elementary, three essential ingredients come into play. The first one is the usual infinite Ramsey theorem, due to Ramsey. The second one is due to El-Zahar and Sauer and was already mentioned when dealing with S 3 . As for the last one, it is due to Milliken (For more on this theorem and its applications, see [START_REF] Todorcevic | Introduction to Ramsey spaces[END_REF]).

The case S = {1, 2, 3, 4} is consequently the only case with S = 4 for which the indivisibility problem remains unsolved. In the present case, it is a bit ironical as {1, 2, 3, 4} is precisely the distance set in which we were interested. So far, the reason for which S 4 stands apart is still unclear. However, it might be that it is actually the very first case were metricity comes into play. Indeed, for all the other sets S with |S| 4, the space U S can be coded as an object where the metric aspect does not appear and this is what makes Ramsey, Milliken and El-Zahar -Sauer theorems helpful. Our feeling is consequently that solving the indivisibility problem for S 4 requires a new approach. Still, we have to admit that what we are hoping for is a positive answer and that Theorem 5 is undoubtedly responsible for that.

3.2. 1/m-indivisibility of the S m 's. We now turn to 1/m-indivisibility of the spaces S m . In Theorem 5, we showed how an exact indivisibility result transfers to an approximate one. It turns out that a slight modification of the proof allows to show: Proposition 7. Assume that for some strictly positive m ∈ ω, S m is indivisible. Then S 3m is 2/3m-indivisible.

Proof. To prove this theorem, it suffices to show that there is an isometric copy S * * m of S m inside S 3m such that for every S m ⊂ S * * m isometric to S m , ( S m ) 2/3m includes an isometric copy of S 3m . The proof is essentially the same as the proof of Proposition 5 where S * m is constructed except that instead of the metric space X m = (S Q , d S Q m ), one works with (S 3m , d S Q m ). The fact that the approximation can be made up to 2/3m and not 1/m comes from the fact that for α ∈ [0, 1] 3m , α ⌈α⌉ m α + 2/3m whereas if α ∈ [0, 1] ∩ Q, one only has α ⌈α⌉ m < α + 1/m. Thus: Theorem 6. For every m 9, S m is 2/m-indivisible.

3.3. Bounds. We now turn to the computation of values ε with respect to which S is ε-indivisible.

Theorem (Theorem 2). S is 1/3-indivisible.
Note also that if at some point an approximate indivisibility result for S m showed up independently of an exact one, we would still be able to compute a bound for S: Proposition 8. Suppose that for some strictly positive integer m, S m is 1/mindivisible. Then S is ε-indivisible for every ε 3/2m. It follows that ( ←χ {i}) ε includes an isometric copy of S.

Concluding remarks and open problems.

The equivalence provided by Theorem 4 suggests several lines of future investigation. Apparently, here is the first and most reasonable question to consider: Question. Is S 4 indivisible? More generally, is S m indivisible for every strictly positive integer m?

We finish with two results which might be useful for that purpose. The first one makes a reference to the space S Q : Proposition 9. Let m ∈ ω be strictly positive. Assume that for every strictly positive k ∈ ω and χ : S Q -→ k, there is a copy S m of S m in S Q on which χ is constant. Then S m is indivisible.

Proof. Once again, we work with X m = (S Q , d S Q m ) and the identity map π m : S Q -→ S m . Think of X m as a subspace of S m . Now, let k ∈ ω be strictly positive and χ : S m -→ k. Then χ induces a coloring of X m , and therefore a coloring χ • π of S Q . By hypothesis, there is a copy S m of S m in S Q on which χ • π is constant with value i < k. Then π ′′ S m ⊂ ←χ {i}. The result follows since π ′′ S m is isometric to S m .

The second result provides a space whose indivisibility is equivalent to the indivisibility of S m . Let P denote the Cantor space, that is the topological product space 2 ω . Let C(P ) denote the set of all continuous maps from P to R equipped with the . ∞ norm. Since the work of Banach and Mazur, it is known that C(P ) is a universal separable metric space. Actually, Sierpinski's proof of that fact allows to show the following result. For m ∈ ω strictly positive, let C m denote the space of all continuous maps from P to [0, 1] m equipped with the distance induced by . ∞ .

Proposition 10. C m is a countable metric space and is universal for the class of all countable metric spaces with distances in [0, 1] m .

It follows that S m is indivisible iff C m is. C m being a much more concrete object than S m , studying its indivisibility might be a alternative to solve the indivisibility problem for S m .

Appendix -Proof of Theorem 1.

Unlike the rest of this paper, this section does not specifically deal with the oscillation stability for S and is simply included here for the sake of completeness. Our purpose is to prove Theorem 1 by constructing the required subspace of Y. Let X 0 ⊂ Y be countable and dense. Then, assuming that X n ⊂ Y countable has been constructed, get X n+1 as follows: Consider F the set of all finite subspaces of X n . For F ∈ F, consider the set E n (F) of all Katȇtov maps f over F with values in the set {d Y (x, y) : x, y ∈ X n } and such that F ∪ {f } embeds into Y. Observe that X n being countable, so are {d Y (x, y) : x, y ∈ X n } and E n (F). Then, for F ∈ F, f ∈ E n (F), fix y f F ∈ Y realizing f over F. Finally, let X n+1 be the subspace of Y with underlying set X n ∪ {y f F : F ∈ F, f ∈ E n (F)}. After ω steps, set X = n∈ω X n . X is clearly a countable dense subspace of Y, and it is ultrahomogeneous thanks to the equivalent formulation of ultrahomogeneity provided in lemma 1.

A second proof involves logical methods. Fix a countable elementary submodel M ≺ H θ for some large enough θ and such that Y, d Y ∈ M . Let X = M ∩ Y. We claim that X has the required property. First, observe that X is dense inside Y since by the elementarity of M , there is a countable D ∈ M (and therefore D ⊂ M ) which is a dense subset of Y. For ultrahomogeneity, let F ⊂ X be finite and let f be a Katȇtov map over F such that F ∪ {f } embeds into X. Observe that f ∈ M . Indeed, dom(f ) ∈ M . On the other hand, let F ∪ {y} ⊂ X be isometric to F ∪ {f } via an isometry ϕ. Then for every x ∈ F, d Y (ϕ(x), y) ∈ M . But d Y (ϕ(x), y) = f (x). Thus, ran(f ) ∈ M . It follows that f is an element of

Lemma 4 .Lemma 5 .

 45 Every non-trivial irreducible cycle in X m has size 3.Proof. Obvious since δ induces the metric d Xm on X m . Every non-trivial irreducible cycle in T has size 3 and is included in a branch.

Figure 1 .

 1 Figure 1. Irreducible cycles

Theorem 5 .

 5 Let S be finite subset of ]0, +∞[ of size |S| 4 and satisfying the 4-values condition. Assume that S ≁ {1, 2, 3, 4}. Then U S is indivisible.

Proof.

  Let ε 3/2m. Consider S * m constructed in Proposition 5. Now, let k ∈ ω be strictly positive and χ : S -→ k. χ induces a coloring of S * m and S m being 1/m-indivisible, there are i < k and S m ⊂ S * m isometric to S m such that S m ⊂ ( ←χ {i}) 1/m . By construction, ( S m ) 1/2m includes an isometric copy of S. Now, (( ←χ {i}) 1/m ) 1/2m ⊂ ( ←χ {i}) 3/2m ⊂ ( ←χ {i}) ε .

  1/m . 2.4. From 1/2(m 2 + m)-indivisibility of S 2(m 2 +m) to indivisibility of S m . We now turn to the proof of the implication (iii) → (iv) of Theorem 4 stating that if for every strictly positive m ∈ ω, S m is 1/m-indivisible, then for every strictly positive m ∈ ω, S m is indivisible. This is done via the following proposition: Proposition 4. Suppose that for some strictly positive integer m, S 2(m 2 +m) is 1/2(m 2 + m)-indivisible. Then S m is indivisible.Proof. Let m ∈ ω be strictly positive and such that S 2(m 2 +m) is 1/2(m 2 + m)indivisible. We are going to create a metric space W with distances in [0, 1] m and a bijection π : S 2(m 2 +m) -→ W such that for every subspace Y of S 2(m 2 +m) , if (Y) 1/2(m 2 +m) includes a copy of S m , then so does π ′′ Y.

M . Now, by ultrahomogeneity of Y, there is y in Y realizing f over F. So by elementarity, there is x in X realizing f over F.