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metric space; Hrushovski property; extension property for partial automorphisms; homogeneous structure; amalgamation class
We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.
[1] Aranda A., Bradley-Williams D., Hubička J., Karamanlis M., Kompatscher M., Konečný M., Pawliuk M.: Ramsey Expansions of Metrically Homogeneous Graphs. available at arXiv:1707.02612 [math.CO], 2017.
[2] Evans D., Hubička J., Konečný M., Nešetřil J.: EPPA for two-graphs and antipodal metric spaces. available at arXiv:1812.11157 [math.CO] (2018), 13 pages.
[3] Evans D. M., Hubička J., Nešetřil J.: Ramsey properties and extending partial automorphisms for classes of finite structures. available at arXiv:1705.02379 [math.CO] (2017), 33 pages.
[4] Hall M. Jr.: Coset representations in free groups. Trans. Amer. Math. Soc. 67 (1949), 421–432. DOI 10.1090/S0002-9947-1949-0032642-4 | MR 0032642
[5] Herwig B., Lascar D.: Extending partial automorphisms and the profinite topology on free groups. Trans. Amer. Math. Soc. 352 (2000), no. 5, 1985–2021. DOI 10.1090/S0002-9947-99-02374-0 | MR 1621745
[6] Hodkinson I.: Finite model property for guarded fragments. 2012, slides available at
[7] Hodkinson I., Otto M.: Finite conformal hypergraph covers and Gaifman cliques in finite structures. Bull. Symbolic Logic 9 (2003), no. 3, 387–405. DOI 10.2178/bsl/1058448678 | MR 2005955
[8] Huang J., Pawliuk M., Sabok M., Wise D.: The Hrushovski property for hypertournaments and profinite topologies. available at arXiv:1809.06435 [math.LO] (2018), 20 pages.
[9] Hubička J., Konečný M., Nešetřil J.: Conant's generalised metric spaces are Ramsey. available at arXiv:1710.04690 [math.CO] (2017), 22 pages.
[10] Hubička J., Konečný M., Nešetřil J.: All those EPPA classes (Strengthenings of the Herwig–Lascar theorem). available at arXiv:1902.03855 [math.CO] (2019), 27 pages.
[11] Hubička J., Nešetřil J.: All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms). available at arXiv:1606.07979 [math.CO] (2016), 59 pages.
[12] Hubička J., Nešetřil J.: Ramsey theorem for designs. The Ninth European Conf. on Combinatorics, Graph Theory and Applications (EuroComb 2017), Viena, 2017, Electronic Notes in Discrete Mathematics 61 (2017), 623–629.
[13] Konečný M.: Semigroup-valued Metric Spaces. Master thesis in preparation available at arXiv:1810.08963 [math.CO], 2018.
[14] Mackey G. W.: Ergodic theory and virtual groups. Math. Ann. 166 (1966), no. 3, 187–207. DOI 10.1007/BF01361167 | MR 0201562
[15] Nešetřil J.: Metric spaces are Ramsey. European J. Comb. 28 (2007), no. 1, 457–468. DOI 10.1016/j.ejc.2004.11.003 | MR 2261831
[16] Nešetřil J., Rödl V.: A structural generalization of the Ramsey theorem. Bull. Amer. Math. Soc. 83 (1977), no. 1, 127–128. DOI 10.1090/S0002-9904-1977-14212-2 | MR 0422035
[17] Otto M.: Amalgamation and symmetry: From local to global consistency in the finite. available at arXiv:1709.00031 [math.CO] (2017), 49 pages.
[18] Pestov V. G.: A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space. Topology Appl. 155 (2008), no. 14, 1561–1575. DOI 10.1016/j.topol.2008.03.002 | MR 2435149
[19] Rosendal Ch.: Finitely approximate groups and actions. Part I: The Ribes-Zalesskiĭ property. J. Symbolic Logic 76 (2011), no. 4, 1297–1306. DOI 10.2178/jsl/1318338850 | MR 2895386
[20] Ribes L., Zalesskii P. A.: On the profinite topology on a free group. Bull. London Math. Soc. 25 (1993), no. 1, 37–43. DOI 10.1112/blms/25.1.37 | MR 1190361
[21] Sabok M.: Automatic continuity for isometry groups. J. Inst. Math. Jussieu (online 2017), 30 pages. MR 3936642
[22] Siniora D., Solecki S.: Coherent extension of partial automorphisms, free amalgamation, and automorphism groups. available at arXiv:1705.01888v3 [math.LO] (2018), 29 pages.
[23] Solecki S.: Extending partial isometries. Israel J. Math. 150 (2005), no. 1, 315–331. DOI 10.1007/BF02762385 | MR 2255813
[24] Solecki S.: Notes on a strengthening of the Herwig–Lascar extension theorem. available at${\scriptstyle\mathtt \sim}$ssolecki/papers/HervLascfin.pdf (2009), 16 pages.
[25] Vershik A. M.: Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space. Topology Appl. 155 (2008), no. 14, 1618–1626. DOI 10.1016/j.topol.2008.03.007 | MR 2435153
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