The Urysohn sphere is oscillation stable

THE UR YSOHN SPHERE IS OSCILLA TION ST ABLE. L. NGUYEN V AN TH ´ E AND N. W. SAUER Abstract. W e solv e the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for ` 2 in the context of the Urysohn space U . This is achiev ed b y solving a purely combinatorial problem in v olving a family of coun table ultrahomogeneous metric spaces with finitely many dis- tances. 1. Introduction. The purp ose of this article is to provide a combinatorial solution for an analog of the distortion problem for ` 2 . This latter problem can b e form ulated as follo ws: let S ∞ denote the unit sphere of the Hilbert space ` 2 . Is it true that if ε > 0 and f : S ∞ − → R is uniformly contin uous, then there is a closed infinite-dimensional subspace V of ` 2 suc h that sup {| f ( x ) − f ( y ) | : x, y ∈ V ∩ S ∞ } < ε ? Equiv alen tly , for a metric space X = ( X , d X ), a subset Y ⊂ X and ε > 0, let ( Y ) ε = { x ∈ X : ∃ y ∈ Y d X ( x, y ) 6 ε } . Then the distortion problem for ` 2 asks: given a finite partition γ of S ∞ , is there alw ays Γ ∈ γ such that (Γ) ε includes V ∩ S ∞ for some closed infinite-dimensional subspace V of ` 2 ? That problem app eared in the early seven ties when Milman’s w ork led to the following prop ert y , which is at the heart of Dvoretzky’s theorem: Theorem (Milman [27]) . L et γ b e a finite p artition of S ∞ . Then for every ε > 0 and every N ∈ N , ther e is Γ ∈ γ and an N -dimensional subsp ac e V of ` 2 such that V ∩ S ∞ ⊂ (Γ) ε . In that con text, the distortion problem for ` 2 really asks whether this result has an infinite dimensional analog. It is only a long time after Milman’s theorem w as established that the distortion problem for ` 2 w as solv ed by Odell and Schlumprec h t in [29]: Theorem (Odell-Sc hlumprec ht [29]) . Ther e is a finite p artition γ of S ∞ and ε > 0 such that no (Γ) ε for Γ ∈ γ includes V ∩ S ∞ for any close d infinite-dimensional subsp ac e V of ` 2 . This result is traditionally stated in terms of the Banach space theoretic concept of oscillation stability , but can also b e stated thanks to a new concept of oscillation Date : April, 2007. 2000 Mathematics Subje ct Classific ation. Primary: 03E02. Secondary: 22F05, 05C55, 05D10, 22A05, 51F99. Key wor ds and phr ases. T opological groups actions, Oscillation stabilit y , Ramsey theory , Met- ric geometry , Urysohn metric space. 1 stabilit y for top ological groups in tro duced by Kechris, P estov and T o dorcevic in [21] (cf [31] for a detailed exp osition). In this latter formalism, the theorem of Odell and Schlumprec ht is equiv alent to the fact that the standard action of iso( S ∞ ) on S ∞ is not oscillation stable. On the other hand, in th e context of isometry groups of complete separable ultrahomogeneous metric spaces, oscillation stability for top ological groups coincides with the Ramsey-theoretic concept of appr oximate indivisibility . Recall that a metric space is called ultr ahomo gene ous when every isometry b etw een finite metric subspaces of X can b e extended to an isometry of X onto itself. F or ε > 0, call a metric space X ε -indivisible when for every finite partition γ of X , there is Γ ∈ γ and e X ⊂ X isometric to X such that e X ⊂ (Γ) ε . Then X is appr oximately indivisible when X is ε -indivisible for every ε > 0, and X is indivisible when X is 0-indivisible. Using this terminology , the theorem of Odell and Schlumprec ht states that the sphere S ∞ is not approximately indivisible. How ever, b ecause the pro of is not based on the in trinsic geometry of ` 2 , the impression somehow p ersists that something is still missing in our understanding of the metric structure of S ∞ . That fact was one of the motiv ations for [25] as well as for the presen t paper: our hope is that understanding the indivisibility problem for another remark able space, namely the Urysohn sphere S , will help to reach a b etter grasp of S ∞ . The space S is defined as follows: up to isometry , it is the unique complete separable ultrahomogeneous metric space with diameter 1 into which ev ery separable metric space with diameter less or equal to 1 embeds isometrically . Equiv alen tly , it is also the sphere of radius 1 / 2 in the so-called universal Urysohn space U , a space to whic h it is closely related. The story of S is quite uncommon: like U , it w as constructed in the late t wen ties by Urysohn (hence quite early in the history of metric geometry) but was completely forgotten for a long time. It is only recently that it was brought back on the researc h scene, thanks in particular to the work of Katˇ eto v [20] which was quickly follo wed b y several results due to Uspenskij [42], [43] and later supp orted by sev eral con tributions b y V ershik [44], [45], Gromov [15], Pesto v [30] and Bogat yi [1], [2]. T o day , the spaces U and S are ob jects of active research and are studied by many differen t authors under many different asp ects, see [34]. Apart from the fact that b oth S ∞ and S are complete, separable and ultraho- mogeneous, the study of S is b eliev ed to b e relev an t for the distortion problem for ` 2 b ecause, from a Ramsey-theoretic p oint of view, the spaces S ∞ and S b eha ve in a very similar wa y . F or example, the following analog of Milman’s theorem holds for S : Theorem (Pesto v [30]) . L et γ b e a finite p artition of S . Then for every ε > 0 and every c omp act K ⊂ S , ther e is Γ ∈ γ and an isometric c opy e K of K in S such that e K ⊂ (Γ) ε . In fact, since the work of Gromo v and Milman [16] and of Pesto v [30], it is kno wn that this analogy is only the most elementary form of a v ery general Ramsey- theoretic theorem. It is also known that this latter result has a very elegant refor- m ulation at the level of the surjective isometry groups iso( S ∞ ) and iso( S ) (seen as top ological groups when equipp ed with the p oint wise conv ergence topology). Call a top ological group G extr emely amenable when every contin uous action of G on a compact space admits a fixed p oint. Then on the one hand: 2 Theorem (Gromov-Milman [16]) . The gr oup iso( S ∞ ) is extr emely amenable. While on the other hand: Theorem (Pesto v [30]) . The gr oup iso( S ) is extr emely amenable. Actually , ev en more is kno wn as both iso( S ∞ ) and iso( S ) are known to satisfy the so-called L´ evy pr op erty (cf Gromo v-Milman [16] for iso( S ∞ ) and Pesto v [32] for iso( S )), a prop erty sho wn to b e stronger than extreme amenability b y Gromov and Milman in [16]. In this note, we prov e that: Theorem 1. The Urysohn spher e S is appr oximately indivisible. In other w ords, for every finite partition γ of S and ε > 0, there is Γ ∈ γ suc h that (Γ) ε includes an isometric cop y of S . Or equiv alently , in terms of oscillation stability for topological groups, the standard action of iso( S ) on S is oscillation stable. Theorem 1 therefore exhibits an essential Ramsey-theoretic distinction b etw een S ∞ and S . A t the level of iso( S ∞ ) and iso( S ), it answers a question mentioned b y Kec hris, Pesto v and T o dorcevic in [21], Hjorth in [17] and Pesto v in [31], and highligh ts a deep topological difference whic h, for the reasons mentioned previously , w as not at all apparent until now. Our pro of of Theorem 1 is combinatorial and rests on a discretization metho d largely inspired from the pro of by Gow ers of the stabilization theorem for the unit sphere S c 0 of c 0 and its p ositive part S + c 0 . Recall that c 0 is the space of all real sequences conv erging to 0 equipp ed with the k·k ∞ norm, and that S + c 0 is the set of all those elemen ts of S c 0 taking only p ositiv e v alues. In [14], Go wers studied the indivisibilit y prop erties of the spaces FIN m (resp. FIN + m ) of all the elements of S c 0 taking only v alues in { k /m : k ∈ [ − m, m ] ∩ Z } (resp. { k /m : k ∈ { 0 , 1 , . . . , m }} ) where m ranges ov er the strictly p ositive integers: Theorem (Gow ers [14]) . L et m ∈ N , m > 1 . Then FIN m (r esp. FIN + m ) is 1 - indivisible (r esp. indivisible). A strong form of these results (see [14] for the precise statement) then led to: Theorem (Gow e rs [14]) . The spher e S c 0 (r esp. S + c 0 ) is appr oximately indivisible. Here, our pro of builds on the following discretization result pro ved in [25] and in volving a family ( U m ) m > 1 of countable metric spaces. F or m > 1, the space U m is defined as follows: up to isometry it is the unique countable ultrahomogeneous metric space with distances in { 1 , . . . , m } into whic h every countable metric space with distances in { 1 , . . . , m } embeds isometrically . Then: Theorem (Lop ez-Abad - Nguyen V an Th´ e [25]) . The fol lowing ar e e quivalent: (i) The sp ac e S is appr oximately indivisible. (ii) F or every strictly p ositive m ∈ N , U m is indivisible. The main ideas of the implication ( ii ) → ( i ) are presented for completeness in section 5 together with an explanation as of why the spaces U m are relev ant as w ell as why some of the previous attempts to pro ve Theorem 1 failed. F or more details, see the original reference [25] or [28]. In the present pap er, we show: Theorem 2. L et m ∈ N , m > 1 . Then U m is indivisible. 3 Theorem 2 expands the list of already known partition results of so-called coun t- able ultrahomogeneous relational structures. Those structures app eared in the late fifties thanks to the pioneering w ork of F ra ¨ ıss ´ e [12] and ha ve since b een studied from v arious points of view. This led in particular to several deep combinatorial clas- sification results (see Lac hlan-W o o drow [23] for graphs, Schmerl [39] for partially ordered sets, Cherlin [5] for directed graphs, or more recently Gray-Macpherson [13] for connected-ultrahomogeneous graphs) but also to substantial developmen ts in permutation group theory (e.g. Cameron [3], T russ [40]), logic (e.g. Pouzet-Roux [33]), or Ramsey theory (initiated b y Komj´ ath-R¨ odl [22]). How ever, although our pap er really b elongs to combinatorics, several consequences of Theorem 1 related to functional analysis deserv e to be men tioned. They are based on the follo wing fact, which is easily seen to b e equiv alent to Theorem 1: Theorem 3. L et X b e a sep ar able metric sp ac e with finite diameter δ . Assume that every sep ar able metric sp ac e with diameter less or e qual to δ emb e ds isometric al ly into X . Then X is appr oximately indivisible. When applied to the unit sphere of certain remark able Banach spaces, this theo- rem yields interesting consequences. F or example, it is known that every separable metric space with diameter less or equal to 2 em b eds isometrically into the unit sphere S C [0 , 1] of the Banach space C [0 , 1]. It follo ws that: Theorem 4. The unit spher e of C [0 , 1] is appr oximately indivisible. On the other hand, it is also kno wn that C [0 , 1] is not the only space having a unit sphere satisfying the h yp otheses of Theorem 3. F or example, Holmes prov ed in [18] that there is a Banach space H suc h that for every isometry i : U − → Y of the Urysohn space U into a Banach space Y with 0 Y is in the range of i , there is an isometric isomorphism b etw een H and the closed linear span of i ( U ) in Y . V ery little is kno wn about the space H , but it is easy to see that its unit sphere con tains isometrically ev ery separable metric space with diameter less or equal to 2. Therefore: Theorem 5. The unit spher e of the Holmes sp ac e is appr oximately indivisible. Observ e that these result do not sa y that for X = C [0 , 1] or H , every finite partition γ of the unit sphere S X of X and ev ery ε > 0, there is Γ ∈ γ and a closed infinite dimensional subspace Y of X such that S X ∩ Y ⊂ (Γ) ε : according to the classical results ab out oscillation stability in Banac h spaces, this latter fact is false for those Banach spaces into which every separable Banach space embeds linearly , and it is known that b oth C [0 , 1] and H hav e this prop erty . The pap er is organized as follows. Section 2 corresp onds to a short presentation of the partition theory of countable ultrahomogeneous structures with free amalga- mation. In section 3, the essential ingredien ts, the main technical results (Lemma 2 and Lemma 3) as w ell as the general outline of the pro of of Theorem 2 are pre- sen ted. Finally , the pro of of Lemma 2 is presented in section 4, while section 5 presen ts a brief history of the problem of approximate indivisibility of S together with an outline of the proof of the aforemen tioned result of Lop ez-Abad and the first author. Ac kno wledgements. L. Nguyen V an Th´ e would lik e to ackno wledge the sup- p ort of the Department of Mathematics & Statistics Postdoctoral Program at the 4 Univ ersity of Calgary . N. W. Sauer was supported by NSERC of Canada Grant # 691325. W e would also lik e to thank Jordi Lopez-Abad, Vitali Milman, Stev o T o dorcevic, the mem b ers of the Equip e de Logique set theory group at the Univer- sit y of Paris 7 and the anonymous referee for the considerable impro vemen ts their helpful comments and suggestions brought to the pap er. 2. P ar tition theor y of count able ul trahomogeneous structures with free amalgama tion. In this section, we presen t a brief history of the general theory of indivisibility of coun table ultrahomogeneous relational structures. F or the undefined notions and for a general in tro duction to the partition theory of coun table ultrahomogeneous structures see [35]. As mentioned in the in tro duction, partition theory is one of the asp ects under whic h coun table ultrahomogeneous relational structures were traditionally studied. The pap er [22] quickly follo wed b y [7] initiates a series devoted to this field, and more precisely dev oted to vertex partition results of countable ultrahomogeneous structures with free amalgamation (The partition theory for sets of substructures other than vertices is muc h more complicated, see [24] and [38]). In [8] it is pro ven that if a countable ultrahomogeneous structure is indivisible then the stabilizers of finite subsets form a chain, which in the binary case is a chain under em b edding. This then led to [36] in whic h it is sho wn, in the case of directed graphs, that if the partial order of the stabilizers is finite then the Ramsey degree is equal to the size of its maximal an tichain. In [9] the finiteness condition was remov ed in the case that the partial order is a chain. [37] contains the most general result from whic h it follo ws that the Ramsey degree of a binary countable ultrahomogeneous structure with free amalgamation is equal to the size of the maximal antic hain of the partial order of finite set stabilizers under embedding if this partial order is finite. Hence if this partial order is a chain then the ultrahomogeneous structure is indivisible. F or metric spaces, this global theory do es not apply as amalgamation is in general not free. Still, it allows to capture the most elemen tary cases and manages to giv e a hint of what the general result should b e. Indeed, it is easy to see that the partial order of stabilizers of finite subsets forms a chain under isometric embedding. Moreo ver, if m 6 3, then it can b e noticed that U m has free amalgamation. Hence if m 6 3 then U m is indivisible, a result which allo w ed to prov e that S is 1 / 6- indivisible in [25]. Ho wev er, if m > 3, then the situation c hanges drastically and requires the in- tro duction of essen tially new argumen ts to pro v e that the metric spaces U m are indivisible. The presentation of those arguments is the purpose of the presen t pap er. 3. Not a tions and definitions. In this section, w e present the notions and ob jects that will pla y a cen tral role throughout the pap er. 3.1. Katˇ eto v maps and orbits. Given a metric space X = ( X , d X ), a map f : X − → ]0 , + ∞ [ is Katˇ etov over X when ∀ x, y ∈ X , | f ( x ) − f ( y ) | 6 d X ( x, y ) 6 f ( x ) + f ( y ) . 5 Equiv alently , one can extend the metric d X to X ∪ { f } by defining, for every x, y in X , c d X ( x, f ) = f ( x ) and c d X ( x, y ) = d X ( x, y ). The c orresp onding metric space is then written X ∪ { f } . The set of all Kat ˇ etov maps o ver X is written E ( X ). F or a metric subspace X of Y , a Kat ˇ etov map f ∈ E ( X ) and a p oint y ∈ Y , then y r e alizes f over X if ∀ x ∈ X d Y ( y , x ) = f ( x ) . The set of all y ∈ Y realizing f ov er X is then written O ( f , Y ) and is called the orbit of f in Y . When Y is implied by context, the set O ( f , Y ) is simply written O ( f ). Here, the concepts of Katˇ etov map and orbit are relev ant because of the follo wing standard reformulation of the notion of ultrahomogeneit y , whic h will b e used extensively in the sequel: Lemma 1. L et X b e a c ountable metric sp ac e. Then X is ultr ahomo gene ous iff for every finite subsp ac e F ⊂ X and every Kat ˇ etov map f over F , if F ∪ { f } emb e ds into X , then O ( f , X ) 6 = ∅ . F or a proof of that fact in the general con text of relational structures, see for example [12]. F or a pro of in the particular context of metric spaces, see [28]. Throughout the pap er, we will extensively use the result of Lemma 1 when X = U p , where p > 1 is an integer. Recall that the space U p is defined as follows: it is a coun table, ultrahomogeneous metric space with distances in { 1 , . . . , p } , and ev ery countable metric space with distances in { 1 , . . . , p } embeds isometrically . Moreo ver, it can be pro ved that an y t wo coun table ultrahomogeneous metric spaces with the same finite metric subspaces are isometric (again, this is a standard fact in the context of coun table ultrahomogeneous relational structures, see [12] for a general pro of or [28] for the case of metric spaces). Therefore, the aforementioned prop erties completely characterize U p up to isometry . There are several wa ys to lo ok at U p . F or example, it might b e seen as a very simplified version (as Vladimir Pesto v would sa y , a ”p o or mans version”) of the Urysohn space U already mentioned in the introduction. The space U is, up to isometry , the unique complete separable ultrahomogeneous metric space that is also universal for the class of all separable metric spaces (into whic h an y separable metric space embeds isometrically). The space U was constructed b y Urysohn in [41] whose goal w as precisely to prov e the existence of a universal separable metric space, and there are now adays several known characterizations and constructions of U . F or more information ab out it and its corresponding recent research developmen ts, the reader should refer to the v olume [34]. In the presen t article how ever, it is more imp ortan t to think of the space U p as a discretized version of the Urysohn sphere S (after ha ving replaced the metric d U p b y d U p /p ) whose indivisibilit y prop erties capture the oscillation stability of S , see [25] for the details, or section 5 of the presen t pap er for the main ideas. Remark: The notion of Katˇ eto v function has b ecome standard in the Urysohn space literature b ecause of the construction of U by Kat ˇ etov in [20], often considered as the starting p oin t of the present research ab out U . They app eared prominen tly in sev eral v ery differen t contributions to the field, see for example Cameron-V ershik [4], Melleray [26], Pesto v [30], Uspenskij [43], or V ershik [45]. How ev er, the idea of Kat ˇ eto v function already appears in the original article [41] b y Urysohn and is undoubtedly in the spirit of the constructions provided by F ra ¨ ıss ´ e in [12]. V ery 6 lik ely , as examplified by the referee or by Maurice Pouzet, we are unaw are of many other uses of those ob jects made b y other authors, e.g. Isbell [19] or Flo o d [10], [11]. 3.2. A notion of largeness. In this section, p is a fixed strictly p ositive integer. F or metric spaces X , Y and Z , write X ∼ = Y if there is an isometry from X onto Y and define the set  Z X  as  Z X  = { e X ⊂ Z : e X ∼ = X } . Definition 1. The set P is the set of al l or der e d p airs of the form s = ( f s , C s ) wher e (i) C s ∈  U p U p  . (ii) f s is a map with finite domain dom f s ⊂ C s and with values in { 1 , . . . , p } . (iii) f s ∈ E (dom f s ) , ie f s is Katˇ etov on its domain. The set P is p artial ly or der e d by the r elation 6 define d by ∀ s, t ∈ P t 6 s ↔ (dom f s ⊂ dom f t ⊂ C t ⊂ C s and f t  dom f s = f s ) . Final ly, if k ∈ N , then t 6 k s stands for t 6 s and min f t =  min f s − k if min f s > k , 1 otherwise. Observ e that if s ∈ P , then the ultrahomogeneity of U p ensures that the set O ( f s , C s ) is not empty and isometric to U n where n = min(2 min f s , p ) (indeed, O ( f s , C s ) is coun table ultrahomogeneous with distances in { 1 , . . . , n } and embeds ev ery coun table metric space with distances in { 1 , . . . , n } ). Observe also that there is alwa ys a t ∈ P such that t 6 1 s . Observ e finally that unlike the relations 6 and 6 0 , the relation 6 k is not transitive when k > 0. Definition 2. L et s ∈ P and Γ ⊂ U p . The notion of largeness of Γ relative to s is define d r e cursively as fol lows: If min f s = 1 , then Γ is lar ge r elative to s iff ∀ t 6 0 s ( O ( f t , C t ) ∩ Γ is infinite ) . If min f s > 1 , then Γ is lar ge r elative to s iff ∀ t 6 0 s ∃ u 6 1 t (Γ is lar ge r elative to u ) . The idea b ehind the definition of largeness is that if Γ is large relative to s , then inside C s the set Γ should represent a substantial part of the orbit of f s . This in tuition is made precise by the following Lemma: Lemma 2. L et s ∈ P . Assume that Γ is lar ge r elative to s . Then ther e exists an isometric c opy C of U p inside C s such that: (i) dom f s ⊂ C . (ii) O ( f s , C ) ⊂ Γ . In words, Lemma 2 means that b y thinning up C s , it is p ossible to ensure that the whole orbit of f s is included in Γ. The requiremen t dom f s ⊂ C guarantees that the orbit of f s in the new space has the same metric structure as the orbit of f s in the original space. The pro of of Lemma 2 represen ts the core of the proof 7 of Theorem 2 and is detailed in section 4. The second crucial fact ab out P and largeness lies in: Lemma 3. L et s ∈ P b e such that Γ is not lar ge r elative to s . Then ther e is t 6 0 s such that U p r Γ is lar ge r elative to t . Pr o of. W e pro ceed by induction on min f s . If min f s = 1, then there is t 6 0 s such that O ( f t , C t ) ∩ Γ is finite . It is then clear that U p r Γ is large relativ e to t . On the other hand, if min f s > 1, then there is t 6 0 s such that ∀ w 6 1 t Γ is not large relative to w . W e claim that U p r Γ is large relative to t : let u 6 0 t . W e wan t to find v 6 1 u suc h that U p r Γ is large relative to v . Let w 6 1 u . Then w 6 1 t and it follo ws that Γ is not large relativ e to w . By induction hypothesis, since min f w < min f u = min f t there is v 6 0 w suc h that U p r Γ is large relative to v . Additionally v 6 1 u . Thus v is as required.  When com bined, Lemma 2 and Lemma 3 lead to Theorem 2 as follo ws: T ake p = m and consider a finite partition γ of U m . Without loss of generality , γ has only tw o parts, namely Π (purple p oints) and Ω (orange p oin ts). Fix t ∈ P such that min f t = m . According to Lemma 3, either Π is large relativ e to t or there is u 6 0 s such that Ω is large relativ e to u . In any case, there are s ∈ { t, u } and Γ ∈ { Π , Ω } such that min f s = m and Γ is large relative to s . Applying Lemma 2 to s , we obtain a copy C of U m inside C s suc h that dom f s ⊂ C and O ( f s , C ) ⊂ Γ. Observ e that O ( f s , C ) is isometric to U m .  The remaining part of this article is therefore devoted to a pro of of Lemma 2. 4. Proof of Lemma 2. F rom no w on, the integer p > 0 is fixed together with Γ ⊂ U p . W e pro ceed by induction and prov e that for every strictly positive m ∈ N with m 6 p the following statemen t J m holds: J m : ”F or every s ∈ P such that min f s = m , if Γ is large relativ e to s , then there exists an isometric copy C of U p inside C s suc h that: (i) dom f s ⊂ C . (ii) O ( f s , C ) ⊂ Γ.” This section is organized as follows. In subsection 4.1, w e show that the state- men t J m is equiv alent to a stronger statemen t denoted H m . This is achiev ed thanks to a technical lemma (Lemma 5) ab out the structure of the orbits in U p and whose pro of is postp oned to subsection 4.5. In subsection 4.2, we initiate the proof by induction and show that the statement J 1 holds. W e then show that if H j holds for ev ery j < m , then J m holds. The general strategy of the induction step is presen ted in subsection 4.3, while 4.4 provides the details for the most technical asp ects. 8 4.1. Reform ulation of J m . As men tioned previously , we start by reformulating the statemen t J m under a form whic h will b e useful when p erforming the induction step. F or a function f and a subset F of the domain dom f of f , we write f  F for the restriction of f to F . Consider the following statement, denoted H m : H m : ”F or every s ∈ P and every F ⊂ dom f s suc h that min f s  F = min f s = m , if Γ is large relative to s , then there exists an isometric copy C of U p inside C s suc h that: (i) dom f s ∩ C = F . (ii) O ( f s  F , C ) ⊂ Γ.” The statement J m is clearly implied by H m : simply take F = dom f s . The purp ose of the following lemma is to show that the conv erse is also true. Lemma 4. The statement J m implies the statement H m . Pr o of. Our main to ol here is the following technical result, whose proof is p ostp oned to section 4.5. Lemma 5. L et G 0 ⊂ G b e finite subsets of U p , G a family of Katˇ etov maps with domain G and such that for every g , g 0 ∈ G : max( | g − g 0 |  G 0 ) = max | g − g 0 | , min(( g + g 0 )  G 0 ) = min( g + g 0 ) . Then ther e exists an isometric c opy C of U p inside U p such that: (i) G ∩ C = G 0 . (ii) ∀ g ∈ G O ( g  G 0 , C ) ⊂ O ( g , U p ) . Note that under the conditions of Lemma 5, the restriction map g 7→ g  G 0 is one-to-one. Assuming Lemma 5, here is how J m implies H m : let s and F be as in the hypothesis of H m . Apply J m to s to get an isometric copy e C of U p inside C s suc h that dom f s ⊂ e C and O ( f s , e C ) ⊂ Γ. Apply then Lemma 5 inside e C to F ⊂ dom f s and the family { f s } to get an isometric cop y C of U p inside e C such that dom f s ∩ C = F and O ( f s  F , C ) ⊂ O ( f s , e C ). Then C is as required.  4.2. Pro of of J 1 . Consider an enumeration { x n : n ∈ N } of C s admitting dom f s as an initial segment. Assume that the p oints ϕ ( x 0 ) , . . . , ϕ ( x n ) are constructed so that: • The map ϕ is an isometry . • ϕ  dom f s = id dom f s . • ϕ ( x k ) ∈ Γ whenever ϕ ( x k ) realizes f s o ver dom f s . W e wan t to construct ϕ ( x n +1 ). Consider h defined on { ϕ ( x k ) : k 6 n } by: ∀ k 6 n h ( ϕ ( x k )) = d C s ( x k , x n +1 ) . Observ e that the metric subspace of C s giv en by { x k : k 6 n + 1 } witnesses that h is Kat ˇ etov. It follo ws that the set of all y ∈ C s r dom f s realizing h ov er { ϕ ( x k ) : k 6 n } is not empt y and ϕ ( x n +1 ) can b e chosen in that set. Additionally , observ e that if h  dom f s = f s , then the fact that min f s = 1 and Γ is large relative to s then guarantees that h can b e realized b y a point in Γ. W e can therefore c ho ose ϕ ( x n +1 ) to b e one of those p oints. After infinitely man y steps, the subspace C of C s supp orted by { ϕ ( x n ) : n ∈ N } is as required.  9 4.3. Induction step. Assume that the statements J 1 . . . J m − 1 , and therefore the statemen ts H 1 . . . H m − 1 hold. W e are going to sho w that J m holds. So let s ∈ P suc h that min f s = m and Γ is large relative to s . T o make the notation easier, we assume that s is of the form ( f , U p ) and we write F instead of dom f . W e need to pro duce an isometric copy C of U p inside U p suc h that F ⊂ C and O ( f , C ) ⊂ Γ. This is achiev ed inductiv ely thanks to the follo wing lemma. Recall that for metric subspaces X and Y of U p and ε > 0, the sets ( X ) ε and  Y U p  are defined by: ( X ) ε = { y ∈ U p : ∃ x ∈ X d U p ( y , x ) 6 ε } ,  Y U p  = { e U ⊂ Y : e U ∼ = U p } . Lemma 6. L et X b e a finite subsp ac e of U p and A ∈  U p U p  such that: (i) F ⊂ X ⊂ A . (ii) ( X ) m − 1 ∩ O ( f , A ) ⊂ Γ . (iii) ∀ g ∈ E ( X ) ( g  F = f  F ) → (Γ is lar ge r elative to ( g , A )) . Then for every h ∈ E ( X ) , ther e ar e B ∈  A U p  and x ∗ ∈ B r e alizing h over X such that: (i’) F ⊂ ( X ∪ { x ∗ } ) ⊂ B . (ii’) ( X ∪ { x ∗ } ) m − 1 ∩ O ( f , B ) ⊂ Γ . (iii’) ∀ g ∈ E ( X ∪ { x ∗ } ) ( g  F = f  F ) → (Γ is lar ge r elative to ( g , B )) . Claim 1. L emma 6 implies J m . Pr o of. The required copy of C can b e constructed inductiv ely . W e start by fixing an en umeration { x n : n ∈ N } of U p suc h that F = { x 0 , . . . , x k } and by setting ˜ x i = x i for every i 6 k . Next, we pro ceed as follows: set A k = U p . Then the subspace of U p supp orted by { ˜ x 0 , . . . , ˜ x k } and the cop y A k satisfy the requirements (i)-(iii) of Lemma 6. Consider then h k +1 defined on { ˜ x 0 , . . . , ˜ x k } by: ∀ i 6 k h k +1 ( ˜ x i ) = d U p ( x k +1 , x i ) . Then h k +1 is Katˇ eto v ov er { ˜ x 0 , . . . , ˜ x k } and Lemma 6 can b e applied to the subspace of U p supp orted by { ˜ x 0 , . . . , ˜ x k } , the copy A k and the Kat ˇ eto v map h k +1 . It produces x ∗ and B , and we set ˜ x k +1 = x ∗ and A k +1 = B . In general, assume that ˜ x 0 , . . . , ˜ x l and A k , . . . , A l are constructed so that A l and the subspace of U p supp orted by { ˜ x 0 , . . . , ˜ x l } satisfy the hypotheses of Lemma 6. Consider h l +1 defined on { ˜ x 0 , . . . , ˜ x l } by: ∀ i 6 l h l +1 ( ˜ x i ) = d U p ( x l +1 , x i ) . Then h l +1 is Kat ˇ eto v ov er { ˜ x 0 , . . . , ˜ x l } , Lemma 6 can b e applied to pro duce x ∗ and B , and we set ˜ x l +1 = x ∗ and A l +1 = B . After infinitely steps, w e are left with C = { ˜ x n : n ∈ N } isometric to U p , as required.  The remaining part of this section is consequen tly dev oted to a pro of of Lemma 6 where X , A and h are fixed according to the requirements (i)-(iii) of Lemma 6. Claim 2. If x ∗ and B satisfy (i’) and (ii’) of L emma 6, then (iii’) is also satisfie d. 10 Pr o of. Let g ∈ E ( X ∪ { x ∗ } ) b e such that g  F = f  F . W e need to show that Γ is large relativ e to ( g , B ). If min g > m , then ( g , B ) 6 0 ( f , U p ). Since Γ is large relativ e to ( f , U p ), it follows that Γ is also large relative to ( g , B ) and we are done. On the other hand, if min g 6 m − 1, then O ( g , B ) ⊂  ( X ∪ { x ∗ } ) m − 1 ∩ O ( f , B )  ⊂ Γ . So Γ is large relative to ( g , B ).  With this fact in mind, we define K = { φ ∈ E ( X ∪ { h } ) : φ  F = f  F and φ ( h ) 6 m − 1 } . Tw o commen ts about notation b efore w e go on. First, as sp ecified in 3.1, X ∪ { h } in the definition of K ab ov e is understo o d as the one-point metric extension X ∪ { h } of X obtained by setting d ( x, h ) = h ( x ) and d ( x, y ) = d X ( x, y ) for every x, y in X . Next, in the sequel, when X ⊂ U p and X ∪ { u } , X ∪ { v } are one-p oint metric extensions X (provided by p oints of U p r X or by Katˇ etov maps ov er X ), w e will write X ∪ { u } ∼ = X ∪ { v } when d ( x, u ) = d ( x, v ) whenev er x ∈ X . The reason for whic h K is relev an t here lies in the following claim. Claim 3. Assume that B ∈  A U p  and x ∗ ∈ B ar e such that: (i) X ⊂ B . (ii) x ∗ r e alizes h over X . (iii) F or every φ ∈ K , every p oint in B r e alizing φ over X ∪ { x ∗ } ∼ = X ∪ { h } is in Γ . (iv) x ∗ ∈ Γ if h  F = f  F (that is if x ∗ ∈ O ( f , B ) ). Then x ∗ and B satisfy (i’) and (ii’) L emma 6. Pr o of. The requirement (i’) is obviously satisfied so w e concentrate on (ii’). Let y ∈ ( X ∪ { x ∗ } ) m − 1 ∩ O ( f , B ). W e need to prov e that y ∈ Γ. If y ∈ ( X ) m − 1 , then y is actually in ( X ) m − 1 ∩ O ( f , A ) ⊂ Γ and w e are done. Otherwise, y ∈ ( { x ∗ } ) m − 1 . If y = x ∗ , there is nothing to do: since y is in O ( f , B ), so is x ∗ . Th us, by (iv), x ∗ ∈ Γ, that is y ∈ Γ. Otherwise, let φ b e the Katˇ etov map realized b y y ov er X ∪ { x ∗ } ∼ = X ∪ { h } . According to (iii), it suffices to show that φ ∈ K . This is what w e do now. First, the metric space X ∪ { x ∗ , y } witnesses that φ is Kat ˇ eto v o v er X ∪ { h } . Next, y ∈ O ( f , B ) hence φ  F = f  F . Finally , φ ( h ) = d U p ( x ∗ , y ) 6 m − 1 since y ∈ ( { x ∗ } ) m − 1 .  The strategy to construct B and x ∗ is the following one. Let { φ α : α < | K |} b e an en umeration of K . W e first construct a sequence of p oints ( x α ) α< | K | and a decreasing sequence ( D α ) α< | K | of copies of U p so that x α ∈ D α and for ev ery β 6 α < | K | : (i) X ⊂ D α . (ii) x α realizes h ov er X . (iii) Ev ery p oint in D α realizing φ β o ver X ∪ { x α } ∼ = X ∪ { h } is in Γ. The details of this construction are pro vided in section 4.4. Once this is done, call x 0 = x | K |− 1 , B 0 = D | K |− 1 . The p oin t x 0 and the cop y B 0 are almost as required except that x 0 ma y not be in Γ. If h  F 6 = f  F , this is not a problem and setting x ∗ = x 0 and B = B 0 w orks. On the other hand, if h  F = f  F , then some extra w ork is required and we pro ceed as follows. 11 Pic k x ∗ ∈ B 0 realizing h ov er X and suc h that d U p ( x ∗ , x 0 ) = 1. W e will b e done if w e construct B ∈  B 0 U p  so that ( X ∪ { x ∗ , x 0 } ) ∩ B = X ∪ { x ∗ } and for every φ ∈ K , ev ery p oin t in B realizing φ ov er X ∗ ∪ { x ∗ } realizes φ ov er X ∗ ∪ { x 0 } . Here is ho w this is achiev ed thanks to Lemma 5. F or φ ∈ K , define the map ˆ φ on X ∪ { x ∗ , x 0 } b y  ˆ φ  X = φ  X , ˆ φ ( x ∗ ) = ˆ φ ( x 0 ) = φ ( h ) . Using the fact that φ is Kat ˇ etov o ver X ∪ { h } and X ∪ { x ∗ } ∼ = X ∪ { x 0 } ∼ = X ∪ { h } , it is easy to chec k that ˆ φ is Katˇ etov ov er X ∪ { x ∗ , x 0 } and that for every φ, φ 0 ∈ K : max( | ˆ φ − ˆ φ 0 |  X ∪ { x ∗ } ) = max | ˆ φ − ˆ φ 0 | , min(( ˆ φ + ˆ φ 0 )  X ∪ { x ∗ } ) = min( ˆ φ + ˆ φ 0 ) . W orking inside B 0 , we can therefore apply Lemma 5 to X ∪ { x ∗ } ⊂ X ∪ { x ∗ , x 0 } and the family ( ˆ φ ) φ ∈ K to obtain B as required.  4.4. Construction of the sequences ( x α ) α< | K | and ( D α ) α< | K | . The construc- tion of the sequences ( x α ) α< | K | and ( D α ) α< | K | is carried out thanks to a rep eated application of the following lemma. Recall that the set K of Katˇ eto v functions ov er X ∪ { h } is defined by K = { φ ∈ E ( X ∪ { h } ) : φ  F = f  F and φ ( h ) 6 m − 1 } . Note that when X ∪ { u } ∼ = X ∪ { h } , we will often see K as a set of Kat ˇ etov maps ov er X ∪ { u } . Every element of K is then thought of as a Katˇ etov map ov er X ∪ { u } in the obvious manner. Lemma 7. L et F ⊂ K and D ∈  A U p  b e such that X ⊂ D . Assume that u ∈ D r e alizes h over X and is such that for every φ ∈ F , every p oint in D r e alizing φ over X ∪ { u } ∼ = X ∪ { h } is in Γ . L et s ∈ K r F b e such that ∀ φ ∈ K φ ( h ) > s ( h ) → φ ∈ F and φ ( h ) < s ( h ) → φ / ∈ F . ( ∗ ) Then ther e ar e E ∈  D U p  and v ∈ E r e alizing h over X such that X ⊂ E and for every φ ∈ F ∪ { s } , every p oint in E r e alizing φ over X ∪ { v } ∼ = X ∪ { h } is in Γ . Once Lemma 7 is pro ven, here is ho w the sequences ( x α ) α< | K | and ( D α ) α< | K | are constructed: choose the enumeration { φ α : α < | K |} of K so that the sequence ( φ α ( h )) α< | K | is nondecreasing. Apply Lemma 7 to F = ∅ , D = A and s = φ 0 to pro duce x 0 and D 0 . In general, apply Lemma 7 to F = { φ 0 . . . φ α } , D = D α and s = φ α +1 to produce x α +1 and D α +1 . After | K | steps, the sequences ( x α ) α< | K | and ( D α ) α< | K | are as required. Pr o of of L emma 7. W e start with the case where s ( h ) > min s  X . The map s b eing in K , s ( h ) 6 m − 1 and so min s  X 6 m − 1. Then, O ( s  X , D ) ⊂  ( X ) m − 1 ∩ O ( f , D )  . But from the requirement (ii) of Lemma 6,  ( X ) m − 1 ∩ O ( f , D )  ⊂ Γ . Observ e no w that every p oint in D realizing s o ver X ∪ { u } is in O ( s  X , D ). Th us, according to the previous inclusions, any such p oint is also in Γ. So in fact, there is nothing to do: v = u and E = D works. 12 F rom now on, we consequen tly supp ose that s ( h ) < min s  X . Let s 1 b e defined on X ∪ { u } by s 1 ( x ) =  s ( x ) if x ∈ X , s ( h ) + 1 if x = u . Claim 4. The map s 1 is Katˇ etov. Pr o of. The map s is Kat ˇ eto v ov er X . Hence, it is enough to prov e that for every x ∈ X , | s 1 ( u ) − s 1 ( x ) | 6 d U p ( x, u ) 6 s 1 ( u ) + s 1 ( x ) . That is | s ( h ) + 1 − s ( x ) | 6 h ( x ) 6 s ( h ) + 1 + s ( x ) . Because s is Katˇ etov o ver X ∪ { h } , it is enough to prov e that s ( h ) + 1 − s ( x ) 6 h ( x ) . But this holds since s ( h ) < min s  X .  Note that, as p ointed out by the referee, the previous claim also admits a nice geometric explanation: proving that s 1 is Kat ˇ etov is equiv alent to v erifying that the metric space X ∪ { u, s } stays metric when the distance betw een u and s is increased b y one. T o do that, simply observe that any metric triangle with integer distances (in particular, here, those of the form { x, u, s } ) remains metric when a distance that is not the largest is increased by one (which is true here b ecause s ( h ) < min s  X ). Claim 5. Γ is lar ge r elative to ( s 1 , D ) . Pr o of. If s ( h ) = m − 1, then min s 1 = m = min f and so ( s 1 , D ) 6 0 ( f , U p ). Since Γ is large relativ e to ( f , U p ), it is also large relativ e to ( s 1 , D ) and we are done. On the other hand, if s ( h ) < m − 1, then s 1 ∈ K and it follo ws from the h yp othesis ( ∗ ) on F that s 1 ∈ F . In particular, every p oint in D realizing s 1 o ver X ∪ { u } is in Γ, and it follows that Γ is large relative to ( s 1 , D ).  Consequen tly , there is ( s 2 , D s 2 ) 6 1 ( s 1 , D ) such that Γ is large relative to ( s 2 , D s 2 ). W e are now going to construct v and a Katˇ eto v extension s 3 of s 2 suc h that v realizes h o ver X , s 3 ( v ) = s ( h ) and ( s 3 , D s 2 ) 6 0 ( s 2 , D s 2 ). This last requiremen t will make sure that Γ is large relativ e to ( s 3 , D s 2 ). W e will then apply Lemma 5 to obtain the cop y E as required. Here is ho w we pro ceed formally: fix w ∈ O ( s 2 , D s 2 ) and consider the map h 1 defined on X ∪ { u, w } by h 1 ( x ) =    h ( x ) if x ∈ X . 1 if x = u . s ( h ) if x = w . Claim 6. The map h 1 is Katˇ etov. Pr o of. The metric space ( X ∪ { h } ) ∪ { s } witnesses that h 1  X ∪ { w } is Kat ˇ eto v. Next, h 1  X ∪ { u } is also Katˇ etov: Let x ∈ X . Then | h 1 ( x ) − h 1 ( u ) | = h ( x ) − 1 6 h ( x ) = d U p ( x, u ) 6 h ( x ) + 1 = h 1 ( x ) + h 1 ( u ) . The only thing we still need to show is therefore | h 1 ( u ) − h 1 ( w ) | 6 d U p ( u, w ) 6 h 1 ( u ) + h 1 ( w ) . But this inequalities hold as they are equiv alent to | 1 − s ( h ) | 6 s ( h ) + 1 6 1 + s ( h ) .  13 Let v ∈ D s 2 realizing h 1 o ver X ∪ { u, w } . As announced previously , define an extension s 3 of s 2 on dom s 2 ∪ { v } by setting s 3 ( v ) = s ( h ). Claim 7. The map s 3 is Katˇ etov and Γ is lar ge r elative to ( s 3 , D s 2 ) . Pr o of. The point w realizes s 3 o ver dom s 2 ∪ { v } and therefore witnesses that s 3 is Katˇ etov. As for Γ, it is large relative to ( s 3 , D s 2 ) b ecause it is large relativ e to ( s 2 , D s 2 ) and ( s 3 , D s 2 ) 6 0 ( s 2 , D s 2 ).  Observ e now that min s 3 = s ( h ) = min s 3  X ∪ { u, v } = min s 6 m − 1. Thus, one can apply H min s inside D s 2 to s 3 and X ∪ { u, v } to obtain D s 3 ∈  D s 2 U p  suc h that dom s 3 ∩ D s 3 = X ∪ { u, v } and O ( s 3  X ∪ { u, v } , D s 3 ) ⊂ Γ. At that p oint, b oth u and v realize h ov er X and if φ ∈ F , then ev ery p oint in D s 3 realizing φ ov er X ∪ { u } is in Γ. Thus, we will b e done if we can construct E ∈  D s 3 U p  suc h that: • ( X ∪ { u, v } ) ∩ E = X ∪ { v } . • F or every φ ∈ F , every p oint in E realizing φ ov er X ∪ { v } realizes φ o ver X ∪ { u } . • Ev ery p oint in E realizing s ov er X ∪ { v } realizes s 3 o ver X ∪ { u, v } . Once again, this is ac hieved thanks to Lemma 5: for φ ∈ F , define the map ˆ φ on X ∪ { u, v } by:  ˆ φ  X = φ  X , ˆ φ ( u ) = ˆ φ ( v ) = φ ( h ) . Using the fact that φ is Katˇ etov o ver X ∪ { h } and X ∪ { u } ∼ = X ∪ { v } ∼ = X ∪ { h } , it is easy to chec k that ˆ φ is Kat ˇ etov o ver X ∪ { u, v } . Let b F = ( ˆ φ ) φ ∈F . W orking inside D s 3 , we w ould like to apply Lemma 5 to X ∪ { v } ⊂ X ∪ { u, v } and the family { s 3 } ∪ b F to obtain E as required. It is therefore enough to c heck: Claim 8. F or every g , g 0 ∈ { s 3 } ∪ b F : max( | g − g 0 |  X ∪ { v } ) = max | g − g 0 | , min(( g + g 0 )  X ∪ { v } ) = min( g + g 0 ) . Pr o of. When g , g 0 ∈ b F , this is easily done. W e therefore concentrate on the case where g = ˆ φ for φ ∈ F and g 0 = s 3 . What we hav e to do is to show that: | ˆ φ ( u ) − s 3 ( u ) | 6 max( | ˆ φ − s 3 |  X ∪ { v } ) (1) ˆ φ ( u ) + s 3 ( u ) > min(( ˆ φ + s 3 )  X ∪ { v } ) (2) Recall first that s 3 ( u ) = s ( h ) + 1 and that s 3 ( v ) = s ( h ). Remember also that according to the prop erties of F , s ( h ) 6 φ ( h ). F or (1), if s ( h ) < φ ( h ), then we are done since | ˆ φ ( u ) − s 3 ( u ) | = | φ ( h ) − ( s ( h ) + 1) | = φ ( h ) − ( s ( h ) + 1) 6 φ ( h ) − s ( h ) = φ ( v ) − s 3 ( v ) 6 | ˆ φ ( v ) − s 3 ( v ) | . On the other hand, if φ ( h ) = s ( h ), then | ˆ φ ( u ) − s 3 ( u ) | = 1 but then this is less than or equal to max( | ˆ φ − s 3 |  X ∪ { v } ) as this latter quan tity is equal to max | φ − s | , 14 whic h is at least 1 since φ ∈ F and s / ∈ F . Thus, the inequality (1) holds. As for (2), simply observe that ˆ φ ( u ) + s 3 ( u ) > ˆ φ ( v ) + s 3 ( v ) .  This finishes the pro of of Lemma 7.  4.5. Pro of of Lemma 5. The purp ose of this section is to provide a pro of of Lemma 5 whic h was used extensively in the previous pro ofs. Let G 0 ⊂ G b e finite subsets of U p , G a family of Katˇ eto v maps with domain G and such that for every g , g 0 ∈ G : max( | g − g 0 |  G 0 ) = max | g − g 0 | , min(( g + g 0 )  G 0 ) = min( g + g 0 ) . W e need to pro duce an isometric copy C of U p inside U p suc h that: (i) G ∩ C = G 0 . (ii) ∀ g ∈ G O ( g  G 0 , C ) ⊂ O ( g , U p ) . First, observe that it suffices to provide the pro of assuming that G is of the form G 0 ∪ { z } . The general case is then handled by rep eating the pro cedure. Lemma 8. L et X b e a finite subsp ac e of S { O ( g  G 0 ) : g ∈ G } . Then ther e is an isometry ϕ on U p fixing G 0 ∪ ( X ∩ S { O ( g ) : g ∈ G } ) and such that: ∀ g ∈ G ϕ ( X ∩ O ( g  G 0 )) ⊂ O ( g ) . Pr o of. F or x ∈ X , there is a unique element g x ∈ G such that x ∈ O ( g x  G 0 ). Let k be the map defined on G 0 ∪ X by k ( x ) =  d U p ( x, z ) if x ∈ G 0 , g x ( z ) if x ∈ X . Claim 9. The map k is Katˇ etov. Pr o of. The metric space G 0 ∪ { z } witnesses that k is Katˇ eto v ov er G 0 . Hence, it suffices to chec k that for every x ∈ X and y ∈ G 0 ∪ X , | k ( x ) − k ( y ) | 6 d U p ( x, y ) 6 k ( x ) + k ( y ) . Consider first the case y ∈ G 0 . Then d U ( x, y ) = g x ( y ) and we need to c heck that | g x ( z ) − d U p ( y , z ) | 6 g x ( y ) 6 g x ( z ) + d U p ( y , z ) . Or equiv alently , | g x ( z ) − g x ( y ) | 6 d U p ( y , z ) 6 g x ( z ) + g x ( y ) . But this is true since g x is Katˇ etov ov er G 0 ∪ { z } . Consider now the case y ∈ X . Then k ( y ) = g y ( z ) and we need to chec k | g x ( z ) − g y ( z ) | 6 d U p ( x, y ) 6 g x ( z ) + g y ( z ) . But since X is a subspace of S { O ( g  G 0 ) : g ∈ G } , we hav e, for every u ∈ G 0 , | d U p ( x, u ) − d U p ( u, y ) | 6 d U p ( x, y ) 6 d U p ( x, u ) + d U p ( x, u ) . Since x ∈ O ( g x  G 0 ) and y ∈ O ( g y  G 0 ), this is equiv alent to | g x ( u ) − g y ( u ) | 6 d U p ( x, y ) 6 g x ( u ) + g y ( u ) . Therefore, max( | g x − g y |  G 0 ) 6 d U p ( x, y ) 6 min(( g x + g y )  G 0 ) . 15 No w, b y hypothesis on G , this latter inequalit y remains v alid if G 0 is replaced b y G 0 ∪ { z } . The required inequality follows.  By ultrahomogeneit y of U p (or, more precisely , by its equiv alent reformulation pro vided in Lemma 1), we can consequently realize the map k ov er G 0 ∪ X b y a p oint z 0 ∈ U p . The metric space G 0 ∪ ( X ∩ S { O ( g ) : g ∈ G } ) ∪ { k } b eing isometric to the subspace of U p supp orted b y G 0 ∪ ( X ∩ S { O ( g ) : g ∈ G } ) ∪ { z } , so is the subspace of U p supp orted by G 0 ∪ ( X ∩ S { O ( g ) : g ∈ G } ) ∪ { z 0 } . By ultrahomogeneit y again, we can therefore find a surjective isometry ϕ of U p fixing G 0 ∪ ( X ∩ S { O ( g ) : g ∈ G } ) and such that ϕ ( z 0 ) = z . Then ϕ is as required: let g ∈ G and x ∈ O ( g  G 0 ). Then: d U p ( ϕ ( x ) , z ) = d U p ( ϕ ( x ) , ϕ ( z 0 )) = d U p ( x, z 0 ) = k ( x ) = g ( z ) . That is, ϕ ( x ) ∈ O ( g ).  Lemma 9. Ther e is an isometric emb e dding ψ of G 0 ∪ S { O ( g  G 0 ) : g ∈ G ) } into G 0 ∪ S { O ( g ) : g ∈ G ) } fixing G 0 such that: ∀ g ∈ G ψ ( O ( g  G 0 )) ⊂ O ( g ) . Pr o of. Let { x n : n ∈ N } enumerate S { O ( g  G 0 ) : g ∈ G ) } . F or n ∈ N , let g n b e the only g ∈ G such that x n ∈ O ( g n  G 0 ). Apply Lemma 8 inductively to construct a sequence ( ψ n ) n ∈ N of surjective isometries of U p suc h that for every n ∈ N , ψ n fixes G 0 ∪ ψ n − 1 ( { x k : k < n } ) and ψ n ( x n ) ∈ O ( g n ). Then ψ defined on G 0 ∪ { x n : n ∈ N } b y ψ  G 0 = id G 0 and ψ ( x n ) = ψ n ( x n ) is as required.  W e now turn to the pro of of Lemma 5. Let Y and Z be the metric subspaces of U p supp orted by G ∪ S { O ( g ) : g ∈ G ) } and G 0 ∪ S { O ( g  G 0 ) : g ∈ G ) } resp ectiv ely . Let i 0 : Z − → U p b e the isometric em b edding pro vided b y the iden tity . By Lemma 9, the space Z em b eds isometrically into Y via an isometry j 0 that fixes G 0 . W e can therefore consider the metric space W obtained by gluing U p and Y via an iden tification of Z ⊂ U p and j 0 ( Z ) ⊂ Y . The space W is described in Figure 1. F ormally , the space W can b e constructed thanks to a property of countable met- ric spaces with distances in { 1 , . . . , p } known as str ong amalgamation : we can find a countable metric space W with distances in { 1 , . . . , p } and isometric embeddings i 1 : U p − → W and j 1 : Y − → W such that: • i 1 ◦ i 0 = j 1 ◦ j 0 . • W = i 1 ( U p ) ∪ j 1 ( Y ). • i 1 ( U p ) ∩ j 1 ( Y ) = ( i 1 ◦ i 0 ) ( Z ) = ( j 1 ◦ j 0 ) ( Z ). • F or every x ∈ U p and y ∈ Y : d W ( i 1 ( x ) , j 1 ( y )) = min { d W ( i 1 ( x ) , i 1 ◦ i 0 ( z )) + d W ( j 1 ◦ j 0 ( z ) , j 1 ( y )) : z ∈ Z } = min { d U p ( x, i 0 ( z )) + d Y ( j 0 ( z ) , y ) : z ∈ Z } = min { d U p ( x, z ) + d Y ( j 0 ( z ) , y ) : z ∈ Z } . The crucial p oint here is that in W , every x ∈ i 1 ( U p ) realizing some g  G 0 o ver i 1 ( G 0 ) also realizes g ov er j 1 ( G ). Using W , we show how C can b e constructed inductiv ely: Consider an enumer- ation { x n : n ∈ N } of i 1 ( U p ) admitting i 1 ( G 0 ) as an initial segment. Assume that the p oints ϕ ( x 0 ) , . . . , ϕ ( x n ) are constructed so that: • The map ϕ is an isometry . 16 U p G 0 O ( g 1  G 0 ) O ( g 2  G 0 ) O ( g 3  G 0 ) j 0 G 0 G O ( g 1 ) O ( g 2 ) O ( g 3 ) i 1 i 0 i 1 ( U p ) W j 1 j 1 ( G ) Figure 1. The space W • dom ϕ ⊂ i 1 ( U p ). • ran ϕ ⊂ U p . • ϕ ( i 1 ( x )) = x whenever x ∈ G 0 . • d U p ( ϕ ( x k ) , z ) = d W ( x k , j 1 ( z )) whenever z ∈ G and k 6 n . W e wan t to construct ϕ ( x n +1 ). Consider e defined on { ϕ ( x k ) : k 6 n } ∪ G by:  ∀ k 6 n e ( ϕ ( x k )) = d W ( x k , x n +1 ) , ∀ z ∈ G e ( z ) = d W ( j 1 ( z ) , x n +1 ) . Observ e that the metric subspace of W given by { x k : k 6 n + 1 } ∪ j 1 ( G ) witnesses that e is Katˇ etov. It follows that the set E of all y ∈ U p realizing e ov er the set { ϕ ( x k ) : k 6 n } ∪ G is not empty and ϕ ( x n +1 ) can b e chosen in E .  5. Appendix The purp ose of this section is tw ofold. First, it is to provide a brief presentation of the attempts to solve the approximate indivisibility problem for S . Then, it is to giv e a short outline of the pro of from [25] according to which the indivisibility of the spaces U m implies the approximate indivisibility of S . The motiv ation for a combinatorial attac k of the appro ximate indivisibilit y prob- lem for the Urysohn sphere is based on t wo ideas. The first one is that the com- binatorial p oint of view is relev ant for the study of countable ultrahomogeneous metric spaces in general. The second idea is that the complete separable ultra- homogeneous metric spaces are closely linked to the countable ultrahomogeneous 17 metric spaces. This connection is supp orted b y the fact that every complete sep- arable ultrahomogeneous metric space Y includes a countable ultrahomogeneous dense metric subspace (for a pro of, see [25]). F or example, consider the r ational Urysohn sp ac e U Q whic h can b e defined up to isometry as the unique countable ultrahomogeneous metric space with rational distances for which every coun table metric space with rational distances embeds isometrically . The Urysohn space U arises then as the completion of U Q , a fact whic h is actually essential as it is at the heart of several imp ortant con tributions ab out U . In particular, in the original article [41] of Urysohn, the space U is pre- cisely constructed as the completion of U Q whic h is in turn constructed by hand. Similarly , the Urysohn sphere S arises as the completion of the so-called r ational Urysohn spher e S Q , defined up to isometry as the unique countable ultrahomoge- neous metric space with distances in Q ∩ [0 , 1] into which every at most countable metric space with distances in Q ∩ [0 , 1] embeds isometrically . With resp ect to the approximate indivisibility problem, this latter fact natu- rally leads to the question of knowing whether S Q is indivisible. This question was answ ered b y to Delhomm´ e, Laflamme, Pouzet and Sauer in [6], where a detailed analysis of metric indivisibilit y is pro vided and sev eral obstructions to indivisibilit y are isolated. Cardinality is suc h an obstruction: any separable indivisible metric space must b e at most countable. Unboundedness is another example: any indivis- ible metric space must ha ve a b ounded distance set. It turns out that S Q a voids those 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 6 1 such that there is an ε -c hain ( x i ) i 6 n con taining x and such that d X ( x 0 , x n ) > l . Then, define λ ( x ) = inf { λ ε ( x ) : ε > 0 } . Theorem (Delhomm´ e-Laflamme-Pouzet-Sauer [6]) . L et X b e a c ountable metric sp ac e. Assume that ther e is x 0 ∈ X such that λ ( x 0 ) > 0 . Then X is not indivisible. F or S Q , it is easy to see that ultrahomogeneity together with the fact that the distance set con tains 0 as an accumulation p oint imply that every p oint x in S Q is suc h that λ ( x ) = 1. It follo ws that: Corollary (Delhomm´ e-Laflamme-Pouzet-Sauer [6]) . 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 b een indivisible, S w ould ha ve b een oscillation stable. But in the presen t case, the coloring which was used to divide S Q did not lead to any conclusion concerning the approximate indivisibility problem for S . Later, the idea of using the spaces U m simply came from the fact that essen tially , what makes S Q divisible is the richness of its distance set. The hop e was then that b y w orking with those simpler spaces, one may b e able to a void the problem en- coun tered abov e. The results of the presen t paper show that this hope w as justified, but of course, the v ery first step was to sho w that the appro ximate indivisibilit y prop ert y for S could be captured by the indivisiblit y of U m , or, equiv alen tly , by the indivisibilit y of S m = ( U m , d U m /m ). This was one of the purp oses of [25], where the result is achiev ed by proving the following proposition (section 2.5 in [25]): Prop osition. Assume that for some strictly p ositive m ∈ ω , S m is indivisible. Then S is 1 /m -indivisible. 18 Pr o of. This is obtained by sho wing that there is a separable metric space Z with distances in [0 , 1] and including a copy S ∗ m of S m suc h that for every e S m ⊂ S ∗ m isometric to S m , the set ( e S m ) 1 /m includes an isometric copy of S Q . This prop erty indeed suffices to prov e the Prop osition: the space Z is separable with distances in [0 , 1] so by univ ersality of S w e may assume that it is actually a subspace of S . Let no w γ b e a finite partition of S . It induces a finite partition of the copy S ∗ m . By indivisibilit y of S m , find Γ ∈ γ and e S m ⊂ S ∗ m suc h that e S m ⊂ Γ. By construction of Z , the set ( e S m ) 1 /m includes an isometric cop y of S Q . Observ e that since the metric completion of S Q is S , the closure of ( e S m ) 1 /m in S includes a cop y of S , and we are done since ( e S m ) 1 /m is closed in S . T o construct Z , we first construct a metric space Y m defined on the set S Q × { 0 , 1 } and where the metric d Y m satisfies, for every x, y ∈ S Q : • d Y m (( x, 1) , ( y , 1)) = d S Q ( x, y ). • d Y m (( x, 0) , ( y , 0)) =  d S Q ( x, y )  m (the least k /m > d S Q ( x, y )). • d Y m (( x, 0) , ( x, 1)) = 1 /m . The space Y m is really a t wo-lev el metric space with a low er level we call X m . Note that X m em b eds in to S m b ecause it is a countable metric space with distances in { k /m : k ∈ { 0 , . . . , m }} . Note also that in Y m , ( X m ) 1 /m includes a copy of S Q . So the basic idea to construct Z is to start from a cop y of S m , call it S ∗ m , and to use some kind of gluing tec hnique to glue a cop y of Y m on S ∗ m along e X m whenev er e X m is a cop y of X m inside S ∗ m . Because eac h copy of S m con tains a copy of X m , this pro cess adds a copy of S Q inside ( e S m ) 1 /m whenev er e S m ⊂ S ∗ m is isometric to S m . 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