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The Urysohn sphere is oscillation stable. (English) Zbl 1191.46012

A metric space \(X\) is ultrahomogeneous if for all finite \(F,G\subseteq X\) and isometries \(T\) from \(F\) onto \(G\), \(T\) extends to an isometry of \(X\) onto \(X\). The Urysohn sphere \(S\) is the unique (up to isometry) complete separable ultrahomogeneous metric space of diameter 1 into which every separable metric space, of diameter at most 1, embeds isometrically. For \(\varepsilon\geq0\), a metric space \(X\) is \(\varepsilon\)-indivisible if whenever \(\gamma\) is a finite partition of \(X\) there exist \(\Gamma\in\gamma\) and \(\widetilde X\subseteq X\), \(\widetilde X\) isometric to \(X\), so that \(\widetilde X \subseteq (\Gamma)_{\varepsilon} \equiv \{ x\in X : d(x,\Gamma)\leq \varepsilon\}\). \(X\) is approximately indivisible if it is \(\varepsilon\)-indivisible for all \(\varepsilon>0\).
From the main result of [E.Odell and T.Schlumprecht, “The distortion problem”, Acta Math.173, No.2, 259–281 (1994; Zbl 0828.46005)] we have that \(S^\infty\), the unit sphere of \(\ell_2\), is not approximately indivisible.
As the authors point out, \(S\) shares much with \(S^\infty\). The analog of Milman’s theorem for \(\ell_2\) holds: For all finite partitions \(\gamma\) of \(S\), \(\varepsilon>0\), and compact \(D\subseteq S\), there exist \(\Gamma\in \gamma\) and \(\widetilde K\subseteq S\), \(\widetilde K\) isometric to \(K\), with \(\widetilde K\subseteq (\Gamma)_\varepsilon\) [V.Pestov, “Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups”, Isr.J.Math.127, 317–357 (2002; Zbl 1007.43001)]. Moreover, \(\text{iso}(S^\infty)\), the group of onto isometries, is extremely amenable [M.Gromov and V.D.Milman, “A topological application of the isoperimetric inequality”, Am.J.Math.105, 843–854 (1983; Zbl 0522.53039)] as is \(\text{iso}(S)\), cf. V.Pestov [loc.cit.]. Despite the similarity, the main result of this paper is that \(S\) is approximately indivisible. The proof of the theorem is combinatorial and inspired by W.T.Gowers’ result [“Lipschitz functions on classical spaces”, Europ.J.Combinatorics 13, No.3, 141–151 (1992; Zbl 0763.46015)] that the unit sphere of \(c_0\) is approximately indivisible. It also uses an equivalence established in [J.Lopez-Abad and L.N.Van Thé, “The oscillation stability problem for the Urysohn sphere: A combinatorial approach”, Topology Appl.155, No.14, 1516–1530 (2008; Zbl 1149.22020)] that \(S\) is approximately indivisible iff for all \(m\in \mathbb N\), \(U_m\) is 0-indivisible. \(U_m\) is the unique (up to isometry) countable ultrahomogeneous metric space with distances lying in \(\{1,\dots,m\}\) into which every countable metric space with distances lying in \(\{1,\dots,m\}\) embeds isometrically.

MSC:

46B06 Asymptotic theory of Banach spaces
46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
22F05 General theory of group and pseudogroup actions
05C55 Generalized Ramsey theory
05D10 Ramsey theory
22A05 Structure of general topological groups
51F99 Metric geometry
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References:

[1] Bogatyi S.A.: Universal homogeneous rational ultrametric on the space of irrational numbers. Moscow Univ. Math. Bull 55, 20–24 (2000) · Zbl 0984.54039
[2] Bogatyi S.A.: Metrically homogeneous spaces. Russian Math. Surveys 52, 221–240 (2002) · Zbl 1063.54017 · doi:10.1070/RM2002v057n02ABEH000495
[3] P.J. Cameron, Oligomorphic Permutation Groups, London Math. Society Lecture Note Series 152, 1990. · Zbl 0813.20002
[4] Cameron P.J., Vershik A.M.: Some isometry groups of the Urysohn space. Ann. Pure Appl. Logic 143(1-3), 70–78 (2006) · Zbl 1108.54027 · doi:10.1016/j.apal.2005.08.001
[5] Cherlin G.L.: The Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous n-Tournaments, Mem. Amer. Math. Soc. 131, 621 (1998) · Zbl 0978.03029
[6] Delhommé C., Laflamme C., Pouzet M., Sauer N.: Divisibility of countable metric spaces. Europ. J. Combinatorics 28(6), 1746–1769 (2007) · Zbl 1206.54025 · doi:10.1016/j.ejc.2006.06.024
[7] El-Zahar M., Sauer N.W.: The indivisibility of the homogeneous Kn-free graphs. J. Combin. Theory Ser. B 47, 162–170 (1989) · Zbl 0682.05048 · doi:10.1016/0095-8956(89)90017-8
[8] El-Zahar M., Sauer N.W.: On the divisibility of homogeneous hypergraphs. Combinatorica 14, 159–165 (1994) · Zbl 0803.05043 · doi:10.1007/BF01215348
[9] El-Zahar M., Sauer N.W.: Indivisible homogeneous directed graphs and a game for vertex partitions. Discrete Mathematics 291, 99–113 (2005) · Zbl 1058.05032 · doi:10.1016/j.disc.2004.04.023
[10] J. Flood, Free topological vector spaces, PhD Thesis, Australian National University, Canberra, 1975. · Zbl 0545.46052
[11] J. Flood, Free Topological Vector Spaces, Dissertationes Math. (Rozprawy Mat.) 221 (1984). · Zbl 0545.46052
[12] Fraïssé R.: Theory of Relations, Studies in Logic and the Foundations of Mathematics, 145. North-Holland Publishing Co., Amsterdam (2000) · Zbl 0965.03059
[13] Gowers W.T.: Lipschitz functions on classical spaces. Europ. J. Combinatorics 13, 141–151 (1992) · Zbl 0763.46015 · doi:10.1016/0195-6698(92)90020-Z
[14] R. Gray, D. Macpherson, Countable connected-homogeneous graphs, preprint. · Zbl 1274.05212
[15] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser Verlag (1999). · Zbl 0953.53002
[16] Gromov M., Milman V.D.: A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843–854 (1983) · Zbl 0522.53039 · doi:10.2307/2374298
[17] Hjorth G.: An oscillation theorem for groups of isometries, GAFA. Geom. funct. anal. 18, 489–521 (2008) · Zbl 1153.03028 · doi:10.1007/s00039-008-0664-9
[18] Holmes R.: The universal separable metric space of Urysohn and isometric embeddings thereof in Banach spaces. Fund. Math. 140, 199–223 (1992) · Zbl 0772.54022
[19] Isbell J.R.: Six theorems about injective metric spaces. Comment. Math. Helv. 39, 65–76 (1964) · Zbl 0151.30205 · doi:10.1007/BF02566944
[20] M. Katětov, On universal metric spaces, Gen. Topology and its Relations to Modern Analysis and Algebra VI: Proc. Sixth Prague Topol. Symp. 1986 (Z. Frolík, ed.), Heldermann Verlag (1988), 323–330.
[21] Kechris A.S., Pestov V., Todorcevic S.: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, GAFA. Geom. funct. anal. 15, 106–189 (2005) · Zbl 1084.54014 · doi:10.1007/s00039-005-0503-1
[22] Komjáth P., Rödl V.: Coloring of universal graphs. Graphs Combin. 2, 55–60 (1986) · Zbl 0589.05053 · doi:10.1007/BF01788077
[23] Lachlan A.H., Woodrow R.E.: Countable ultrahomogeneous undirected graphs. Trans. Amer. Math. Soc. 262, 51–94 (1980) · Zbl 0471.03025 · doi:10.1090/S0002-9947-1980-0583847-2
[24] Laflamme C., Sauer N., Vuksanovic V.: Canonical partitions of universal structures. Combinatorica 26, 183–205 (2006) · Zbl 1121.03051 · doi:10.1007/s00493-006-0013-2
[25] A. Leiderman, V. Pestov, M. Rubin, S. Solecki, V.V. Uspenskij (eds.), Special Issue: Workshop on the Urysohn space, Ben-Gurion University of the Negev, Beer Sheva, Israel, 21-24 May 2006, Top. Appl. 155:14 (2008), 1451–1634. · Zbl 1151.54303
[26] Lopez-Abad J., NguyenVan Thé L.: The oscillation stability problem for the Urysohn sphere: A combinatorial approach. Topology Appl. 155(14), 1516–1530 (2008) · Zbl 1149.22020 · doi:10.1016/j.topol.2008.03.011
[27] Melleray J.: Stabilizers of closed sets in the Urysohn space. Fund. Math. 189(1), 53–60 (2006) · Zbl 1089.22019 · doi:10.4064/fm189-1-4
[28] Milman V.D.: A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funkcional. Anal. i Priložen. 5(4), 28–37 (1971) (in Russian)
[29] L. Nguyen Van Thé, Théorie de Ramsey structurale des espaces métriques et dynamique topologique des groupes d’isométries, PhD Thesis, Université Paris 7, 2006 (available in English).
[30] Odell E., Schlumprecht T.: The distortion problem. Acta Mathematica 173, 259–281 (1994) · Zbl 0828.46005 · doi:10.1007/BF02398436
[31] Pestov V.: Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups. Israel Journal of Mathematics 127, 317–358 (2002) · Zbl 1007.43001 · doi:10.1007/BF02784537
[32] V. Pestov, Dynamics of Infinite-Dimensional Groups. The Ramsey–Dvoretzky– Milman phenomenon. Revised edition of Dynamics of Infinite-Dimensional Groups and Ramsey-Type Phenomena [Inst. Mat. Pura. Apl. (IMPA), Rio de Janeiro, 2005]. University Lecture Series, 40, American Mathematical Society, Providence, RI, 2006. · Zbl 1123.37003
[33] Pestov V.: The isometry group of the Urysohn space as a Lévy group. Top. Appl. 154(10), 2173–2184 (2007) · Zbl 1127.22001 · doi:10.1016/j.topol.2006.02.010
[34] Pouzet M., Roux B.: Ubiquity in category for metric spaces and transition systems. European J. Combin. 17, 291–307 (1996) · Zbl 0843.03017 · doi:10.1006/eujc.1996.0025
[35] Sauer N.: Partitions of countable homogeneous systems, appendix in the new printing of the book: R. Fraïssé Theory of Relations, Studies in Logic and the Foundations of Mathematics, 145. North-Holland Publishing Co., Amsterdam (2000)
[36] Sauer N.: A Ramsey theorem for countable homogeneous directed graphs. Discrete Mathematics 253, 45–61 (2002) · Zbl 1001.05082 · doi:10.1016/S0012-365X(01)00448-4
[37] Sauer N.: Canonical vertex partitions. Combinatorics Probability and Computing 12, 671–704 (2003) · Zbl 1062.03046 · doi:10.1017/S0963548303005765
[38] Sauer N.: Coloring subgraphs of the Rado graph. Combinatorica 26, 231–253 (2006) · Zbl 1121.03052 · doi:10.1007/s00493-006-0015-0
[39] Schmerl J.H.: Countable homogeneous partially ordered sets. Algebra Universalis 9, 317–321 (1979) · Zbl 0423.06002 · doi:10.1007/BF02488043
[40] Truss J.K.: Generic automorphisms of homogeneous structures. Proc. London Math. Soc. 65, 121–141 (1992) · Zbl 0765.20003 · doi:10.1112/plms/s3-65.1.121
[41] Urysohn P.: Sur un espace m’etrique universel. Bull. Sci. Math. 51, 43–64 (1927) 74-90 · JFM 53.0556.01
[42] Uspenskij V.: On the group of isometries of the Urysohn universal metric space. Comment. Math. Univ. Carolinae 31, 181–182 (1990) · Zbl 0699.54011
[43] Uspenskij V.: On subgroups of minimal topological groups. Topology Appl. 155(14), 1580–1606 (2008) · Zbl 1166.22002 · doi:10.1016/j.topol.2008.03.001
[44] Vershik A.M.: The universal Urysohn space, Gromov’s metric triples, and random metrics on the series of positive numbers. Russian Math. Surveys 53, 921–928 (1998) · Zbl 1005.53036 · doi:10.1070/RM1998v053n05ABEH000069
[45] Vershik A.M.: The universal and random metric spaces. Russian Math. Surveys 356, 65–104 (2004)
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