Logique mathématique
Positive families and Boolean chains of copies of ultrahomogeneous structures
Comptes Rendus. Mathématique, Tome 358 (2020) no. 7, pp. 791-796.

A family of infinite subsets of a countable set X is called positive iff it is closed under supersets and finite changes and contains a co-infinite set. We show that a countable ultrahomogeneous relational structure 𝕏 has the strong amalgamation property iff the set (𝕏)={AX:𝔸𝕏} contains a positive family. In that case the family of large copies of 𝕏 (i.e. copies having infinite intersection with each orbit) is the largest positive family in (𝕏), and for each -embeddable Boolean linear order 𝕃 whose minimum is non-isolated there is a maximal chain isomorphic to 𝕃{min𝕃} in (𝕏),.

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DOI : 10.5802/crmath.82
Classification : 03C15, 03C50, 20M20, 06A06, 06A05
Kurilić, Miloš S. 1 ; Kuzeljević, Boriša 1

1 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad
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Kurilić, Miloš S.; Kuzeljević, Boriša. Positive families and Boolean chains of copies of ultrahomogeneous structures. Comptes Rendus. Mathématique, Tome 358 (2020) no. 7, pp. 791-796. doi : 10.5802/crmath.82. http://www.numdam.org/articles/10.5802/crmath.82/

[1] Ackerman, Nathanael; Freer, Cameron; Patel, Rehana Invariant measures concentrated on countable structures, Forum Math. Sigma, Volume 4 (2016), e17, 59 pages | DOI | MR | Zbl

[2] Cameron, Peter J. Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, 152, Cambridge University Press, 1990, viii+160 pages | DOI | MR | Zbl

[3] Cherlin, Gregory L. The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, Mem. Am. Math. Soc., Volume 131 (1998) no. 621, p. xiv+161 | DOI | MR | Zbl

[4] El-Zahar, Mohamed; Sauer, Norbert W. Ramsey-type properties of relational structures, Discrete Math., Volume 94 (1991) no. 1, pp. 1-10 | DOI | MR | Zbl

[5] Fraïssé, Roland Sur certaines relations qui généralisent l’ordre des nombres rationnels, C. R. Math. Acad. Sci. Paris, Volume 237 (1953), pp. 540-542 | MR | Zbl

[6] Fraïssé, Roland Sur l’extension aux relations de quelques propriétés connues des ordres, C. R. Math. Acad. Sci. Paris, Volume 237 (1953), pp. 508-510 | MR | Zbl

[7] 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

[8] Fraïssé, Roland Theory of relations, Studies in Logic and the Foundations of Mathematics, 145, North-Holland, 2000, ii+451 pages (With an appendix by Norbert Sauer) | MR | Zbl

[9] Kuratowski, Kazimierz Topology. Vol. I, Academic Press Inc.; Państwowe Wydawnictwo Naukowe, 1966, xx+560 pages (New edition, revised and augmented. Translated from the French by J. Jaworowski) | MR

[10] Kurilić, Miloš S. Maximal chains in positive subfamilies of P(ω), Order, Volume 29 (2012) no. 1, pp. 119-129 | DOI | MR | Zbl

[11] Kurilić, Miloš S. Maximal chains of copies of the rational line, Order, Volume 30 (2013) no. 3, pp. 737-748 | DOI | MR | Zbl

[12] Kurilić, Miloš S.; Kuzeljević, Boriša Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs, Adv. Math., Volume 264 (2014), pp. 762-775 | DOI | MR | Zbl

[13] Kurilić, Miloš S.; Kuzeljević, Boriša Maximal chains of isomorphic suborders of countable ultrahomogeneous partial orders, Order, Volume 32 (2015) no. 1, pp. 83-99 | DOI | MR | Zbl

[14] Kurilić, Miloš S.; Kuzeljević, Boriša Antichains of Copies of Ultrahomogeneous Structures, 2019 | arXiv

[15] Macpherson, Dugald A survey of homogeneous structures, Discrete Math., Volume 311 (2011) no. 15, pp. 1599-1634 | DOI | MR | Zbl

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