Fixed points in compactifications and combinatorial counterparts
Annales Henri Lebesgue, Volume 2 (2019), pp. 149-185.


Keywords Ramsey theory, fixed point properties in topological dynamics


The Kechris–Pestov–Todorcevic correspondence connects extreme amenability of topological groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts.


[AH78] Abramson, Fred G.; Harrington, Leo A. Models without indiscernibles, J. Symb. Log., Volume 43 (1978) no. 3, pp. 572-600 | DOI | MR | Zbl

[AKL12] Angel, Omer; Kechris, Alexander S.; Lyons, Russell Random Orderings and Unique Ergodicity of Automorphism Groups, J. Eur. Math. Soc., Volume 16 (2012) no. 10, pp. 2059-2095 | DOI | MR | Zbl

[BJM78] Berglund, John F.; Junghenn, Hugo D.; Milnes, Paul Compact right topological semigroups and generalizations of almost periodicity, Lecture Notes in Mathematics, 663, Springer, 1978, x+243 pages | MR | Zbl

[BK17] Bartošová, Dana; Kwiatkowska, Aleksandra Gowers’ Ramsey theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan, J. Comb. Theory, Ser. A, Volume 150 (2017), pp. 108-136 | DOI | MR | Zbl

[BK18] Bartošová, Dana; Kwiatkowska, Aleksandra The universal minimal flow of the homeomorphism group of the Lelek fan (2018) (to appear in Trans. Am. Math. Soc.) | DOI | Zbl

[BLALM16] Bartošová, Dana; Lopez-Abad, Jordi; Lupini, Martino; Mbombo, Brice R. The Ramsey property for Banach spaces, Choquet simplices, and their noncommutative analogs (2016) ( | Zbl

[Bod15] Bodirsky, Manuel Ramsey classes: examples and constructions, Surveys in combinatorics 2015 (London Mathematical Society Lecture Note Series), Volume 424, Cambridge University Press, 2015, pp. 1-48 | MR | Zbl

[Bou98] Bourbaki, Nicolas Elements of Mathematics. General topology. Chapters 1–4, Springer, 1998, vii+437 pages (Translated from the French, Reprint of the 1989 English translation) | Zbl

[BPT13] Bodirsky, Manuel; Pinsker, Michael; Tsankov, Todor Decidability of definability, J. Symb. Log., Volume 78 (2013) no. 4, pp. 1036-1054 | DOI | MR | Zbl

[BY18] Ben Yaacov, Itaï On a Roelcke-precompact Polish groups that cannot act transitively on a complete metric space, Isr. J. Math., Volume 224 (2018) no. 1, pp. 105-132 | MR | Zbl

[BYMT17] Ben Yaacov, Itaï; Melleray, Julien; Tsankov, Todor Metrizable universal minimal flows of Polish groups have a comeagre orbit, Geom. Funct. Anal., Volume 27 (2017) no. 1, pp. 67-77 | DOI | MR | Zbl

[BYT16] Ben Yaacov, Itaï; Tsankov, Todor Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups, Trans. Am. Math. Soc., Volume 368 (2016) no. 11, pp. 8267-8294 | DOI | MR | Zbl

[dV93] de Vries, Jan Elements of topological dynamics, Mathematics and its Applications, 257, Kluwer Academic Publishers, 1993 | MR | Zbl

[EFH + 16] Eagle, Christophe J.; Farah, Ilijas; Hart, Bradd; Kadets, Boris; Kalashnyk, Vladyslav; Lupini, Martino Fraïssé limits of C * -algebras, J. Symb. Log., Volume 81 (2016) no. 2, pp. 755-773 | DOI | Zbl

[EG17] Etesami, Omid; Ghadernezhad, Zaniar Convex Ramsey matrices and non-amenability of automophism groups of generic structures (2017) (

[EHN16] Evans, David M.; Hubička, Jan; Nešetřil, Jaroslav Automorphism groups and Ramsey properties of sparse graphs (2016) ( | Zbl

[Eng89] Engelking, Ryszard General topology, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, 1989 | Zbl

[Fra54] Fraïssé, Roland Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. Éc. Norm. Supér., Volume 71 (1954), pp. 363-388 | DOI | MR | Zbl

[GKP18] Ghadernezhad, Zaniar; Khalilian, Hamed; Pourmahdian, Massoud Automorphism groups of generic structures: Extreme amenability and amenability, Fundam. Math., Volume 242 (2018) no. 1, pp. 1-23 | DOI | MR | Zbl

[Gla76] Glasner, Shmuel Proximal flows, Lecture Notes in Mathematics, 517, Springer, 1976, viii+153 pages | MR | Zbl

[Gla98] Glasner, Eli On minimal actions of Polish groups, Topology Appl., Volume 85 (1998) no. 1-3, pp. 119-125 | DOI | MR | Zbl

[GLR72] Graham, Roland L.; Leeb, Klaus; Rothschild, Bruce L. Ramsey’s theorem for a class of categories, Adv. Math., Volume 8 (1972), pp. 417-433 | DOI | MR | Zbl

[GLR73] Graham, Roland L.; Leeb, Klaus; Rothschild, Bruce L. Errata: “Ramsey’s theorem for a class of categories”, Adv. Math., Volume 10 (1973), p. 326-327 | MR | Zbl

[GM83] Gromov, Mikhael; Milman, Vitali D. A topological application of the isoperimetric inequality, Am. J. Math., Volume 105 (1983) no. 4, pp. 843-854 | DOI | MR | Zbl

[GM06] Glasner, Eli; Megrelishvili, Michael Hereditarily non-sensitive dynamical systems and linear representations, Colloq. Math., Volume 104 (2006) no. 2, pp. 223-283 | DOI | MR | Zbl

[GM08] Glasner, Eli; Megrelishvili, Michael New algebras of functions on topological groups arising from G-spaces, Fundam. Math., Volume 201 (2008) no. 1, pp. 1-51 | DOI | MR

[GM13] Glasner, Eli; Megrelishvili, Michael Banach representations and affine compactifications of dynamical systems, Asymptotic geometric analysis (Fields Institute Communications), Volume 68, Springer, 2013, pp. 75-144 | DOI | MR | Zbl

[GP07] Giordano, Thierry; Pestov, Vladimir G. Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu, Volume 6 (2007), pp. 279-315 | DOI | MR | Zbl

[GR71] Graham, Roland L.; Rothschild, Bruce L. Ramsey’s theorem for n-parameter sets, Trans. Am. Math. Soc., Volume 159 (1971), pp. 257-292 | MR | Zbl

[Gro52] Grothendieck, Alexander Critères de compacité dans les espaces fonctionnels généraux, Am. J. Math., Volume 74 (1952), pp. 168-186 | DOI | Zbl

[HN16] Hubička, Jan; Nešetřil, Jaroslav All those Ramsey classes (2016) ( | Zbl

[Hod93] Hodges, Wilfrid Model theory, Encyclopedia of Mathematics and Its Applications, 42, Cambridge University Press, 1993, xiii+772 pages | Zbl

[Iba16a] Ibarlucía, Tomas The dynamical hierarchy for Roelcke precompact Polish groups, Isr. J. Math., Volume 215 (2016) no. 2, pp. 965-1009 | DOI | MR | Zbl

[Iba16b] Ibarlucía, Tomas Méthodes de théorie des modèles pour l’étude de groupes topologiques (2016) (Ph. D. Thesis)

[Kir73] Kirillov, Alexandre A. Representations of the infinite-dimensional unitary group, Dokl. Akad. Nauk SSSR, Volume 212 (1973), pp. 288-290 | MR | Zbl

[KPT05] Kechris, Alexander S.; Pestov, Vladimir G.; Todorcevic, Stevo Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal., Volume 15 (2005) no. 1, pp. 106-189 | DOI | Zbl

[KS13] Kubiś, Wiesław; Solecki, Sławomir A proof of uniqueness of the Gurariĭ space, Isr. J. Math., Volume 195 (2013) no. 1, pp. 449-456 | DOI | Zbl

[Kub14] Kubiś, Wiesław Fraïssé sequences: category-theoretic approach to universal homogeneous structures, Ann. Pure Appl. Logic, Volume 165 (2014) no. 11, pp. 1755-1811 | DOI | Zbl

[MNVTT16] Melleray, Julien; Nguyen Van Thé, Lionel; Tsankov, Todor Polish groups with metrizable universal minimal flows, Int. Math. Res. Not. (2016) no. 5, pp. 1285-1307 | DOI | MR | Zbl

[Moo13] Moore, Justin Amenability and Ramsey theory, Fundam. Math., Volume 220 (2013) no. 3, pp. 263-280 | DOI | MR | Zbl

[MT11] Melleray, Julien; Tsankov, Todor Extremely amenable groups via continuous logic (2011) (

[Neš89] Nešetřil, Jaroslav For graphs there are only four types of hereditary Ramsey classes, J. Comb. Theory, Ser. B, Volume 46 (1989) no. 2, pp. 127-132 | DOI | MR | Zbl

[NR77] Nešetřil, Jaroslav; Rödl, Vojtěch Partitions of finite relational and set systems, J. Comb. Theory, Ser. A, Volume 22 (1977) no. 3, pp. 289-312 | DOI | MR | Zbl

[NR83] Nešetřil, Jaroslav; Rödl, Vojtěch Ramsey classes of set systems, J. Comb. Theory, Ser. A, Volume 34 (1983) no. 2, pp. 183-201 | DOI | MR | Zbl

[NVT10] Nguyen Van Thé, Lionel Structural Ramsey theory of metric spaces and topological dynamics of isometry groups, Mem. Am. Math. Soc., Volume 206 (2010) no. 968, p. x+140 | DOI | MR | Zbl

[NVT15] Nguyen Van Thé, Lionel A survey on structural Ramsey theory and topological dynamics with the Kechris–Pestov–Todorcevic correspondence in mind, Zb. Rad. (Beogr.), Volume 17 (2015), pp. 189-207 (volume on Selected topics in combinatorial analysis, updated version available on arXiv) | MR | Zbl

[NVT17] Nguyen Van Thé, Lionel Glasner’s problem for Polish groups with metrizable universal minimal flow (2017) (to appear in Ann. Inst. Fourier) | Zbl

[Pea07] Open problems in topology. II (Pearl, Elliott, ed.), Elsevier, 2007, xii+763 pages | Zbl

[Pes98] Pestov, Vladimir G. On free actions, minimal flows, and a problem by Ellis, Trans. Am. Math. Soc., Volume 350 (1998) no. 10, pp. 4149-4165 | DOI | MR | Zbl

[Pes02] Pestov, Vladimir G. Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Isr. J. Math., Volume 127 (2002), pp. 317-357 | DOI | MR | Zbl

[Pes06] Pestov, Vladimir G. Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon, University Lecture Series, 40, American Mathematical Society, 2006, viii+192 pages | Zbl

[PS16] Pawliuk, Michael; Sokić, Miodrag Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs (2016) ( | Zbl

[RD81] Roelcke, Walter; Dierolf, Susanne Uniform structures on topological groups and their quotients, McGraw-Hill International Book Co., 1981, xi+276 pages | Zbl

[Rup84] Ruppert, Wolfgang Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Mathematics, 1079, Springer, 1984, v+260 pages | DOI | MR | Zbl

[Sol13] Solecki, Sławomir Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem, Adv. Math., Volume 248 (2013), pp. 1156-1198 | DOI | MR | Zbl

[Sol14] Solecki, Sławomir Recent developments in finite Ramsey theory: Foundational aspects and connections with dynamics, Proceedings of the International Congress of Mathematicians, Volume 2 (2014), pp. 103-115 | Zbl

[Tod10] Todorcevic, Stevo Introduction to Ramsey spaces, Annals of Mathematics Studies, 174, Princeton University Press, 2010, vii+287 pages | MR | Zbl

[Tsa12] Tsankov, Todor Unitary representations of oligomorphic groups., Geom. Funct. Anal., Volume 22 (2012) no. 2, pp. 528-555 | DOI | MR | Zbl

[Tsa14] Tsankov, Todor Automorphism groups and their actions (2014) (Habilitation memoir)

[Usp01] Uspenskij, Vladimir V. The Roelcke compactification of groups of homeomorphisms, Topology Appl., Volume 111 (2001) no. 1-2, pp. 195-205 | DOI | MR | Zbl

[Usp02] Uspenskij, Vladimir V. Compactifications of topological groups, Proceedings of the Ninth Prague Topological Symposium (2001) (2002), pp. 331-346 | Zbl

[Usp08] Uspenskij, Vladimir V. On subgroups of minimal topological groups, Topology Appl., Volume 155 (2008) no. 14, pp. 1580-1606 | DOI | MR | Zbl

[Zuc14] Zucker, Andy Amenability and unique ergodicity of automorphism groups of Fraïssé structures, Fundam. Math., Volume 226 (2014) no. 1, pp. 41-62 | DOI | MR | Zbl

[Zuc16] Zucker, Andy Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem, Trans. Am. Math. Soc., Volume 368 (2016) no. 9, pp. 6715-6740 | DOI | MR | Zbl