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Well-quasi-ordering and Embeddability of Relational Structures

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Abstract

At the end of the forties, Fraïssé, following Cantor, Hausdorff and Sierpinski, highlighted the role of the embeddability quasi-order in the theory of relations. Since then, many results illustrating this role have been obtained (a large account was included in Fraïssé’s book Theory of Relations). In this paper, I present a selection of results centered on the notion of well-quasi-order (wqo). I mention several problems on wqo and hereditary classes of relational structures; some of these problems go back to the seventies.

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References

  1. Abdi, D., Laflamme, C., Tateno, A., Woodrow, R.: An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn 42 (2022). arXiv:2205.14679

  2. Albert, M.H., Atkinson, M.D.: Simple permutations and pattern restricted permutations. Discrete Math. 300, 1–15 (2005)

    Article  MathSciNet  Google Scholar 

  3. Albert, M., Brignall, R., Ruškuc, N., Vatter, V.: Rationality for subclasses of 321-avoiding permutations. European J. Comb. 78, 44–72 (2019)

    Article  MathSciNet  Google Scholar 

  4. Allouche, J-P., Shallit, J.: Automatic sequences. Theory, applications, generalizations. Cambridge University Press, Cambridge, xvi+571 (2003)

  5. Assous, R., Pouzet, M.: Jónsson posets. Algebra Universalis 79, Art. 74, 26 (2018)

  6. Atminas, A., Lozin, V.: Labelled induced subgraphs and well-quasi-ordering. Order 32, 313–328 (2015)

    Article  MathSciNet  Google Scholar 

  7. Babai, L.: Graph isomorphism in quasipolynomial time 84 (2016). arXiv:1512.03547

  8. Balogh, J., Bollobás, B., Morris, R.: Hereditary properties of partitions, ordered graphs and ordered hypergraphs. European J. Comb. 8, 1263–1281 (2006)

    Article  MathSciNet  Google Scholar 

  9. Balogh, J., Bollobás, B., Morris, R.: Hereditary properties of ordered graphs. In Topics in discrete mathematics, 179–213, Algorithms Combin., 26, Springer, Berlin, (2006)

  10. Balogh, J., Bollobás, B., Morris, R.: Hereditary properties of combinatorial structures: posets and oriented graphs. J. Graph Theory 56(4), 311–332 (2007)

    Article  MathSciNet  Google Scholar 

  11. Balogh, J., Bollobás, B., Morris, R.: Hereditary properties of tournaments. Electron. J. Combin. 14(1), Research Paper 60, 25 (2007)

  12. Balogh, J., Bollobás, B., Saks, M., Sós, V.T.: The unlabelled speed of a hereditary graph property. J. Combin. Theory Ser B 99, 9–19 (2009)

    Article  MathSciNet  Google Scholar 

  13. Berstel, J., Reutenauer, C.: Noncommutative rational series with applications. Encyclopedia of Mathematics and its Applications, 137. Cambridge University Press, Cambridge, 2011. xiv+248

  14. Bekkali, M., Pouzet, M., Zhani, D.: Incidence structures and Stone-Priestley duality. Ann. Math. Artif. Intell. 49, 27–38 (2007)

    Article  MathSciNet  Google Scholar 

  15. Berge, C.: Les problèmes de flot et de tension. Cah. Cent. Étud. Rech. Opér. 3, 69–93 (1961)

    Google Scholar 

  16. Berstel J. et Reutenauer, C.: Les séries rationnelles et leurs langages. Études et recherches en Informatique. Masson, Paris 132 (1984)

  17. Bonato, A., Tardif, C.: Mutually embeddable graphs and the tree alternative conjecture. J. Combin. Theory Ser. B 96, 874–880 (2006)

    Article  MathSciNet  Google Scholar 

  18. Bonato, A., Bruhn, H., Diestel, R., Sprüssel, P.: Twins of rayless graphs. J. Combin. Theory Ser. B 101, 60–65 (2011)

    Article  MathSciNet  Google Scholar 

  19. Bonnet, E., Giocanti, U., Ossona de Mendez, P., Simon, P., Thomassé, S., Toruńczyk, S.: Twin-width IV:ordered graphs and matrices 53 (2021). arXiv:2102.03117v3

  20. Boudabbous, Y., Pouzet, M.: The morphology of infinite tournaments; applications to the growth of their profile. European J. Combin. 31, 461–481 (2010)

    Article  MathSciNet  Google Scholar 

  21. Braunfeld, S., Laskowski, M.C.: Counting siblings in universal theories 26 (2019). arXiv:1910.11230v3

  22. Brignall, R.: Simplicity in relational structures and its application to permutation classes, PhD Thesis, University of St Andrews, October 25, (2007)

  23. Brignall, R.: Private communication, May 20, (2022)

  24. Brignall, R., Engen, M., Vatter, V.: A counterexample regarding labelled well-quasi-ordering. Graphs Combin. 34, 1395–1409 (2018)

    Article  MathSciNet  Google Scholar 

  25. Brignall, R., Vatter, V.: Labelled well-quasi-order for permutation classes. 48 (2022). arXiv:2103.08243

  26. Brignall, R., Vatter, V.: Uncountably many enumerations of well-quasi-ordered permutation classes, 23 (2023). arXiv:2211.12397v2

  27. Cameron, P.J.: Transitivity of permutation groups on unordered sets. Math. Z. 48, 127–139 (1976)

    Article  MathSciNet  Google Scholar 

  28. Cameron, P.J.: Orbits of permutation groups on unordered sets. J. Lond. Math. Soc. bf 17(2), 410–414 (1978)

    Article  MathSciNet  Google Scholar 

  29. Cameron, P.J.: Orbits of permutation groups on unordered sets. II. J. Lond. Math. Soc. 23(2), 249–264 (1981)

    Article  MathSciNet  Google Scholar 

  30. Cameron, P.J.: Oligomorphic permutation groups. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  31. Cameron, P.J.: The algebra of an age. In: Model theory of groups and automorphism groups (Blaubeuren, 1995), pp. 126–133. Cambridge Univ. Press, Cambridge, (1997)

  32. Cameron, P.J.: Sequences realized by oligomorphic permutation groups. J. Integer Seq. 3, Article 00.1.5 (electronic) (2000)

  33. Cameron, P.J.: Some counting problems related to permutation groups. Formal power series and algebraic combinatorics (Toronto, ON, 1998). Discrete Math. 225, 77–92 (2000)

  34. Cameron, P.J.: On an algebra related to orbit-counting. J. Group Theory 1, 173–179 (1998)

    Article  MathSciNet  Google Scholar 

  35. Cameron, P.J: Homogeneous permutations. Permutation patterns (Otago, 2003). Electron. J. Combin. 9(2), Research paper 2, 9 (2002/03)

  36. Carroy, R., Pequignot, Y.: From well to better, the space of ideals. Fund. Math. 227, 247–270 (2014)

    Article  MathSciNet  Google Scholar 

  37. Chakir, I.: Chains conditions in algebraic lattices. Thèse d’État, 11 Mai 2009, Université Mohamed V, Faculté des Sciences, Rabat (Maroc), 94 (2016). arXiv:1609.07167v1

  38. Chakir, I., Pouzet, M.: A characterization of well-founded algebraic lattices. Contrib. Discrete Math. 13, 35–50 (2018)

    MathSciNet  Google Scholar 

  39. Chakir, I., Pouzet, M.: Infinite independent sets in distributive lattices. Algebra Universalis 53, 211–225 (2005)

    Article  MathSciNet  Google Scholar 

  40. Chudnovsky, M., Kim, R., Oum, S., Seymour, P.: Unavoidable induced subgraphs in large graphs with no homogeneous sets. J. Combin. Theory Ser. B 118, 1–12 (2016)

    Article  MathSciNet  Google Scholar 

  41. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. of Math. 164(2), 51–229 (2006)

    Article  MathSciNet  Google Scholar 

  42. Cohn, P.M.: Universal algebra, Second edition. Mathematics and its Applications, 6. D. Reidel Publishing Co., Dordrecht-Boston, Mass., xv+412 (1981)

  43. Corominas, E.: On better-quasi-ordering countable trees. Discrete Math. 53, 35–53 (1985)

    Article  MathSciNet  Google Scholar 

  44. Courcelle, B., Delhommé, C.: The modular decomposition of countable graphs. Definition and construction in monadic second-order logic. Theoret. Comput. Sci. 394, 1–38 (2008)

  45. Damaschke, P.: Induced subgraphs and well-quasi-ordering. J. Graph Theory 14, 427–435 (1990)

    Article  MathSciNet  Google Scholar 

  46. Dammak, J., Lopez, G., Pouzet, M., Si Kaddour, H.: Hypomorphy of graphs up to complementation. J. Combin. Theory Ser. B 99, 84–96 (2009)

    Article  MathSciNet  Google Scholar 

  47. Davey, B.A., Priestley, H.A.: Introduction to lattices and order, 2nd edn. Cambridge University Press, New York (2002)

    Book  Google Scholar 

  48. Day, G.W.: Superatomic Boolean algebras. Pacific J. Math. 23, 479–489 (1967)

    Article  MathSciNet  Google Scholar 

  49. de Bruijn, N.G., Erdös, P.: A color problem for infinite graphs and a problem in the theory of relations. Indag. Math. 13, 371–373 (1951)

    Article  Google Scholar 

  50. de Jongh, D.H.J., Parikh, R.: Well-partial orderings and hierarchies. Nederl. Akad. Wetensch. Proc. Ser. A 80 (Indag. Math. 39), 195–207 (1977)

  51. Delhommé, C.: Nicely BQO grounded categories and \(2\)-structures, preprint, (2014)

  52. Delhommé, C.: personnal communication

  53. Delhommé, C., Laflamme, C., Pouzet, M., Sauer, N.: Divisibility of countable metric spaces. European J. Combin. 28, 1746–1769 (2007)

    Article  MathSciNet  Google Scholar 

  54. Delhommé, C., Laflamme, C., Pouzet, M., Sauer, N.: Indivisible ultrametric spaces. Topology Appl. 155, 1462–1478 (2008)

    Article  MathSciNet  Google Scholar 

  55. Delhommé, C., Pouzet, M.: Length of an intersection. Mathematical Logic Quarterly 63, 243–255 (2017)

    Article  MathSciNet  Google Scholar 

  56. Delhommé, C., Pouzet, M., Sauer, N., Sági, G.: Representation of ideals of relational structures. Discrete Math. 309, 1374–1384 (2009)

    Article  MathSciNet  Google Scholar 

  57. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math. 51(2), 161–166 (1950)

    Article  MathSciNet  Google Scholar 

  58. Ding, G.: Subgraphs and well-quasi-ordering. J. Graph Theory 16, 489–502 (1992)

    Article  MathSciNet  Google Scholar 

  59. Dushnik, B., Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941)

    Article  MathSciNet  Google Scholar 

  60. Ehrenfeucht, A., Harju, T., Rozenberg, G.: The theory of 2-structures. A framework for decomposition and transformation of graphs. World Scientific Publishing Co., Inc., River Edge, NJ, (1999)

  61. Ehrenfeucht, A., Rozenberg, G.: Theory of 2-structures. I. Clans, basic subclassses, and morphisms. Theoret. Comput. Sci. 70, 277–303 (1990)

  62. Ehrenfeucht, A., Rozenberg, G.: Theory of 2-structures, II. Representation through labeled tree families. Theoret. Comput. Sci. 70, 305–342 (1990)

  63. El-Zahar, M.H., Schmerl, J.H.: On the size of jump-critical ordered sets. Order 1, 3–5 (1984)

    Article  MathSciNet  Google Scholar 

  64. Erdös, P., Tarski, A.: On families of mutually exclusive sets. Ann. of Math. 44(2), 315–329 (1943)

    Article  MathSciNet  Google Scholar 

  65. Erdös, P., Rado, R.: Sets having divisor property, Solution to problem 4358. Amer. Math. Monthly 59, 255–257 (1952)

    MathSciNet  Google Scholar 

  66. Falque, J.: Classification of \(p\)-oligomorphic groups; conjectures of Cameron and Macpherson. Ph.D. thesis, Université Paris-Sud, 136 (2019)

  67. Falque, J., Thiéry, N.M.: Macpherson’s conjecture holds: the orbit algebra of a permutation group with polynomial profile is finitely generated. 30th International conference on formal power series and algebraic combinatorics (FPSAC 2018, Hanover), February (2018)

  68. Fogg, P.: Substitutions in dynamics, arithmetics and combinatorics. V.Berthé, S.Ferenczi, C.Mauduit, A.Siegel (Eds), Springer (2012)

  69. Fraïssé, R.: On a decomposition of relations which generalizes the sum of ordering relations. Bull. Amer. Math. Soc. 59, 389 (1953)

    Google Scholar 

  70. Fraïssé, R.: Sur la comparaison des types de relations. C. R. Acad. Sci. Paris 226, 987–988 (1948)

    MathSciNet  Google Scholar 

  71. Fraïssé, R.: Sur la comparaison des types d’ordres. C. R. Acad. Sci. Paris 226, 1330–1331 (1948)

    MathSciNet  Google Scholar 

  72. Fraïssé, R.: Sur quelques classifications des systèmes de relations. Thèse, Paris (1953). Alger-Math. 1, 35–182 (1954)

  73. Fraïssé, R.: Sur l’extension aux relations de quelques propriétes des ordres. Ann. Sci. Ecole Norm. Sup. 71, 361–388 (1954)

    Google Scholar 

  74. Fraïssé, R.: Abritement entre relations et spécialement entre chaînes, 1971 Symposia Mathematica, Vol. V (INDAM, Rome, 1969/70) pp 203–251 Academic Press, London

  75. Fraïssé, R.: Cours de Logique Mathématique, tome 1, Relation et formule logique, Gauthier-Villars, xiv+179 (1971)

  76. Fraïssé, R.: Theory of relations, Revised edition. With an appendix by Norbert Sauer. Studies in Logic and the Foundations of Mathematics, 145. North-Holland Publishing Co., Amsterdam, ii+451 (2000)

  77. Fraïssé, R.: L’intervalle en théorie des relations, ses généralisations, filtre intervallaires et clôture d’une relation, in "Orders, description and roles", M.Pouzet and D.Richard,éd. (L’Arbresle, 1982), Ann. Discrete Math. 23, 313–341 (1984)

  78. Fraïssé, R., Lopez, G.: La reconstruction d’une relation dans l’hypothèse forte: isomorphie des restrictions à chaque partie stricte de la base. With an appendix by G. Lopez and C. Rauzy. Séminaire de Mathématiques Supérieures, 109. Presses de l’Université de Montréal, Montreal, QC, pp 139 (1990)

  79. Fraïssé, R., Pouzet, M.: Interprétabilité d’une relation par une chaîne. C. R. Acad. Sci. Paris Sér. A-B 272, A1624–A1627 (1971)

  80. Fraïssé, R., Pouzet, M.: Sur une classe de relations n’ayant qu’un nombre fini de bornes. C. R. Acad. Sci. Paris Sér. A-B 273, A275–A278 (1971)

  81. Frasnay, C.: Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes. Thèse. Paris. Annales Institut Fourier Grenoble 15, 415–524 (1965)

    Article  MathSciNet  Google Scholar 

  82. Frasnay, C.: Chainable relations, rangements and pseudorangements, in Orders: description and roles (L’Arbresle, 1982), 235-268, North-Holland Math. Stud., 99, Ann. Discrete Math., 23, North-Holland, Amsterdam, (1984)

  83. Frasnay, C.: Détermination du degré optimal \(d_m\) de monomorphie pour les structures relationnelles au plus m-aires. Math. Rep. Acad. Sci. Canada 12(4), 141–146 (1990)

    MathSciNet  Google Scholar 

  84. Gallai, T.: Transitiv orientbare graphen. Acta Math. Acad. Sci. Hungar. 18, 25–66 (English translation by Maffray, F., Preissmann, M. in Ramirez-Alfonsin, J.J., and Reed, B. (Eds), Perfect graphs, Wiley 2001, 25–66) (1967)

  85. Garey, M.R., Johnson, D.S.: The complexity of near-optimal graph coloring. J. Assoc. Comput. Mach. 23, 43–49 (1976)

    Article  MathSciNet  Google Scholar 

  86. Geniet, C., Thomassé, S.: First order logic and twin-width in tournaments and dense oriented graphs, 28 (2022). arXiv:2207.07683v1

  87. Gibson, P., Pouzet, M., Woodrow, R.: Relational structures having finitely many full-cardinality restrictions. Discrete Math. 291, 115–134 (2005)

    Article  MathSciNet  Google Scholar 

  88. Gottlieb, D.H.: A class of incidence matrices. Proc. Amer. Math. Soc. 17, 1233–1237 (1966)

    Article  MathSciNet  Google Scholar 

  89. Grätzer, G.: General lattice theory, With appendices by Davey, B.A., Freese, R., Ganter, B., Greferath, M., Jipsen, P., Priestley, H.A., Rose, H., Schmidt, E.T., Schmidt, S.E., Wehrung, F., Wille, R., Reprint of the 1998 second edition. Birkhäuser Verlag, Basel, xx+663 (2003)

  90. Hahn, G., Pouzet, M., Woodrow, R.: Siblings of countable cographs, to appear in J. Mult.-Valued Logic Soft Comput. (special issue dedicated to Ivo Rosenberg), 27 (2020). arXiv:2004.12457

  91. Harju, T., Rozenberg, G.: Decomposition of infinite labeled \(2\)-structures. Results and trends in theoretical computer science (Graz, 1994), 145–158. Lecture Notes in Comput. Sci., 812, Springer, Berlin, (1994)

  92. Helfgott, H.A.: Isomorphismes de graphes en temps quasi-polynomial (d’après Babai et Luks, Weisfeiler-Leman), Expository paper associated to Bourbaki seminar (Jan 14, 2017). 43 pages, in French. To appear in Astérisque. Fascicule no 1125 of the Bourbaki seminar (69th year, 2016-2017). arXiv:1701.04372

  93. Hell, P., Nešetřil, J.: Graphs and homomorphisms, Oxford Lecture Series in Mathematics and its Applications, 28. Oxford University Press, Oxford, xii+244 (2004)

  94. Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 2(3), 326–336 (1952)

    Article  MathSciNet  Google Scholar 

  95. Higman, G.: Homogeneous relations. Quart. J. Math. Oxford Ser. 28(2), 31–39 (1977)

    Article  MathSciNet  Google Scholar 

  96. Hodges, W., Lachlan, A.H., Shelah, S.: Possible orderings of an indiscernible sequence. Bull. Lond. Math. Soc. 9, 212–215 (1977)

    Article  MathSciNet  Google Scholar 

  97. Hodkinson, I.M., Macpherson, H.D.: Relational structures determined by their finite induced substructures. J. Symbolic Logic 53, 222–230 (1988)

    Article  MathSciNet  Google Scholar 

  98. Ille, P.: Indecomposable graphs. Discrete Math. 173, 71–78 (1997)

    Article  MathSciNet  Google Scholar 

  99. Jean, M.: Relations monomorphes et classes universelles. C. R. Acad. Sci. Paris Sér. A-B 264, A591–A593 (1967)

  100. Jullien, P.: Contribution á l’étude des types d’ordres dispersés. Thèse Doctorat d’État, Université de Marseille, 27 June (1968)

  101. Kabil, M., Pouzet, M.: Geometric aspects of generalized metric spaces: Relations with graphs, ordered sets and automata in New Trends in Analysis and Geometry, Ali Hussain Alkhaldi, Mohammed Kbiri Alaoui and Mohamed Amine Khamsi (eds), Cambridge Scholars Publishing, 319–377 (2020)

  102. Kantor, W.M.: On incidence matrices of finite projective and affine spaces. Math. Z. 124, 315–318 (1972)

    Article  MathSciNet  Google Scholar 

  103. Kearnes, K., Oman, G.: Jónsson posets and unary Jónsson algebras. Algebra Universalis 69, 101–112 (2013)

    Article  MathSciNet  Google Scholar 

  104. Kechris, A., Pestov, V., Todorcevic, S.: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15, 106–189 (2005)

    Article  MathSciNet  Google Scholar 

  105. Kelly, D.: The 3-irreducible partially ordered sets. Canad. J. Math. 29, 367–383 (1977)

    Article  MathSciNet  Google Scholar 

  106. Kelly, D.: Comparability graphs, in Graphs and Orders, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 147, 3–40 (1985)

  107. Klazar, M.: Overview of general results in combinatorial enumeration, in Permutation patterns. London Math. Soc. Lecture Note Ser. 376, 3–40 (2010)

    Google Scholar 

  108. Korpelainen, N., Lozin, V.: Bipartite induced subgraphs and well-quasi-ordering. J Graph Theory 67, 235–249 (2011)

    Article  MathSciNet  Google Scholar 

  109. Korpelainen, N., Lozin, V., Razgon, I.: Boundary properties of well-quasi-ordered sets of graphs. Order 30, 723–735 (2013)

    Article  MathSciNet  Google Scholar 

  110. Krauss, P.: Universally complete universal theories. Z. Math. Logik Grundlagen Math. 17, 351–370 (1971)

    Article  MathSciNet  Google Scholar 

  111. Kříž, I., Sgall, J.: Well-quasiordering depends on the labels. Acta Sci. Math. (Szeged) 55, 59–65 (1991)

    MathSciNet  Google Scholar 

  112. Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. Amer. Math. Soc. 95, 210–225 (1960)

    MathSciNet  Google Scholar 

  113. Kruskal, J.B.: The theory of well-quasi-ordering: a frequently discovered concept. J. Combin. Theory Ser. A 13, 297–305 (1972)

    Article  MathSciNet  Google Scholar 

  114. Laflamme, C., Pouzet, M., Sauer, N.: Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (Abh. Math. Sem. Univ. Hamburg) October 2017, 87, 2, 369–408 (2015). arXiv:1508.01123

  115. Laflamme, C., Pouzet, M., Sauer, N., Woodrow, R.: Siblings of an \(\aleph _0\)-categorical relational structure. Contrib. Discrete Math. 16(2), 90–127 (2021). Special Edition, Proceedings of the Banff International Workshop 15w5100 Homogeneous Structures (in honour of Norbert Sauer)Invited editor Lionel Nguyen Van Thé, arXiv:1811.04185

  116. Laflamme, C., Pouzet, M., Woodrow, R.: Equimorphy - the case of chains. Arch. Math. Logic 56, 811–829 (2017)

    Article  MathSciNet  Google Scholar 

  117. Laskowski, M.C., Terry, C.A.: Jumps in speeds of hereditary properties in finite relational languages. J. Combin. Theory Ser. B 154, 93–135 (2022)

    Article  MathSciNet  Google Scholar 

  118. Laver, R.: On Fraïssé’s order type conjecture. Ann. of Math. 93(2), 89–111 (1971)

    Article  MathSciNet  Google Scholar 

  119. Laver, R.: An order type decomposition theorem. Ann. of Math. 98(2), 96–119 (1973)

    Article  MathSciNet  Google Scholar 

  120. Lawson, J.D., Mislove, M., Priestley, H.A.: Infinite antichains in semilattices. Order 2, 275–290 (1985)

    Article  MathSciNet  Google Scholar 

  121. Lawson, J.D., Mislove, M., Priestley, H.A.: Ordered sets with no infinite antichains. Discrete Math. 63, 225–230 (1987)

    Article  MathSciNet  Google Scholar 

  122. Lawson, J.D., Mislove, M., Priestley, H.A.: Infinite antichains and duality theories. Houston J. Math. 14, 423–441 (1988)

    MathSciNet  Google Scholar 

  123. Lothaire, M.: Combinatorics on words. Vol. 17 of Encyclopedia of Mathematics and its Applications. Addison-Wesley, Reading, Mass. Reprinted in the Cambridge Mathematical Library, Cambridge University Press, U.K. (1997)

  124. Lozin, V., Gábor, R.: Minimal universal bipartite graphs. Ars Combin. 84, 345–356 (2007)

    MathSciNet  Google Scholar 

  125. Lozin, V., Mayhill, C.: Canonical antichains of unit interval and bipartite permutation graphs. Order 28, 513–522 (2011)

    Article  MathSciNet  Google Scholar 

  126. Macpherson, H.D.: Growth rates in infinite graphs and permutation groups. Proc. Lond. Math. Soc. 51(3), 285–294 (1985)

    Article  MathSciNet  Google Scholar 

  127. Macpherson, H.D.: Infinite permutation groups of rapid growth. J. Lond. Math. Soc. 35(2), 276–286 (1987)

    Article  MathSciNet  Google Scholar 

  128. Macpherson, H.D., Pouzet, M., Woodrow, R.E.: Countable structures of a given age. J. Symb. Log. 57, 992–1010 (1992)

    Article  MathSciNet  Google Scholar 

  129. Mal’cev, A.I.: The metamathematics of algebraic systems, Collected papers: 1936-1967. Translated, edited, and provided with supplementary notes by Benjamin Franklin Wells, III. Studies in Logic and the Foundations of Mathematics, Vol. 66. North-Holland Publishing Co., Amsterdam-London, xviii+494 (1971)

  130. Malliaris, M., Terry, C.: On unavoidable-induced subgraphs in large prime graphs. J. Graph Theory 88, 255–270 (2018)

    Article  MathSciNet  Google Scholar 

  131. Marcone, A.: Foundations of bqo theory. Trans. Amer. Math. Soc. 345, 641–660 (1994)

    Article  MathSciNet  Google Scholar 

  132. Marcus, A., Tardös, G.: Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A 107, 153–160 (2004)

    Article  MathSciNet  Google Scholar 

  133. McKay, G.A.: Better-Quasi-Orders: Extensions and Abstractions. PhD thesis, University of East Anglia, 216 (2015)

  134. Milner, E.C.: Basic wqo- and bqo-theory. in Graphs and order, I.Rival Ed. (Banff, Alta., 1984), 487–502. Reidel, Dordrecht, (1985)

  135. Mislove, M.: When are order scattered and topologically scattered the same?. Orders: description and roles (l’Arbresle, 1982), North Holland Math Stud. 99, Ann. Discrete Math. 23, 61–80 (1984)

  136. Monk, J.D.: Cardinal invariants on Boolean algebras. Progress in Mathematics, vol. 142, Birkhäuser, Basel, (1996)

  137. Moschovakis, Y.N.: Descriptive set theory, Studies in Logic and Foundations of Mathematics, vol. 100, North-Holland Publishing Co., (1980)

  138. On well-quasi-ordering finite trees: Nash-Williams, C. St.J.A. Proc. Cambridge Philos. Soc. 59, 833–835 (1963)

    MathSciNet  Google Scholar 

  139. Nash-Williams, C.St.J.A.: On well-quasi-ordering infinite trees. Proc. Cambridge Philos., Soc. 61, 697–720 (1965)

  140. Oudrar, D.: Sur l’énumération de structures discrètes: une approche par la théorie des relations, Thèse de doctorat, Université d’Alger USTHB à Bab Ezzouar, 249 (2015). arXiv:1604.05839

  141. Oudrar, D.: Hereditary classes of ordered binary structures. Adv. Pure Appl. Math. 11, 1–33 (2020)

    Article  MathSciNet  Google Scholar 

  142. Oudrar, D., Pouzet, M.: Profile and hereditary classes of relational structures. J. Mult.-Valued Logic Soft Comput. 27, 475–500 (2016)

  143. Oudrar, D., Pouzet, M., Zaguia, I.: Minimal prime ages, words and permutation graphs. 46 (2022). arXiv:2206.01557

  144. Oudrar, D., Pouzet, M.: Ordered structures with no finite monomorphic decomposition. Application to the profile of hereditary classes. preprint, 23 (2021)

  145. Pabion, J-F.: Relations préhomogènes. C. R. Acad. Sci. Paris Sér. A-B 274, A529–A531 (1972)

  146. Pouzet, M.: Modèle universel d’une théorie n-complète: Modèle uniformément préhomogène. C. R. Acad. Sci. Paris Sér. A-B 274, A695–A698 (1972)

  147. Pouzet, M.: Un belordre d’abritement et ses rapports avec les bornes d’une multirelation. C. R. Acad. Sci. Paris Sér. A-B 274), A1677–A1680 (1972)

  148. Pouzet, M.: Application d’une propriété combinatoire des parties d’un ensemble aux groupes et aux relations. Math. Z. 150, 117–134 (1976)

    Article  MathSciNet  Google Scholar 

  149. Pouzet, M.: Sur la théorie des relations. Thèse d’état, Université Claude-Bernard, Lyon 1, (1978)

  150. Pouzet, M.: Condition de chaîne en théorie des relations. Israel J. Math. 30, 65–84 (1978)

    Article  MathSciNet  Google Scholar 

  151. Pouzet, M.: Chaînes de théories universelles. Fund. Math. 103, 133–149 (1979)

    Article  MathSciNet  Google Scholar 

  152. Pouzet, M.: Relation minimale pour son âge. Z. Math. Logik Grundlag. Math. 25, 315–344 (1979)

    Article  Google Scholar 

  153. Pouzet, M.: The asymptotic behavior of a class of counting functions. Combinatorics 79 Part 2. M.Deza and I.G.Rosenberg Eds., Ann. Discrete Math. 9, 223–224 (1980)

  154. Pouzet, M.: Relation impartible. Dissertationnes 103, 1–48 (1981)

    Google Scholar 

  155. Pouzet, M.: Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d’une relation. Z. Math. Logik Grundlag. Math. 27, 289–332 (1981)

    Article  Google Scholar 

  156. Pouzet, M.: Applications of well-quasi-ordering and better quasi-ordering. In: Graphs and order (Banff, Alta., 1984), pp. 503–519. Reidel, Dordrecht, (1985)

  157. Pouzet, M.: The profile of relations. Glob. J. Pure Appl. Math. 2, 237–272 (2006)

    MathSciNet  Google Scholar 

  158. Pouzet, M.: When is the orbit algebra of a group an integral domain? Proof of a conjecture of P. J. Cameron. Theor. Inform. Appl. 42, 83–103 (2008)

    Article  MathSciNet  Google Scholar 

  159. Pouzet, M.: Well-quasi-ordering and bounds of hereditary classes of relational structures. in preparation, September (2023)

  160. Pouzet, M., Rosenberg, I.G.: Sperner properties for groups and relations. Europ. J. Combin. 7, 349–370 (1986)

    Article  MathSciNet  Google Scholar 

  161. Pouzet, M., Sauer, N.: From well-quasi-ordered sets to better-quasi-ordered sets. Electron. J. Combin. 13(1), Research Paper 101, 27 (2006)

  162. Pouzet, M., Sobrani, M.: Sandwiches of ages. Ann. Pure Appl. Logic 108, 295–326 (2001)

    Article  MathSciNet  Google Scholar 

  163. Pouzet, M., Sobrani, M.: The order type of the collection of finite series-parallel posets. Discrete Math. 265, 189–211 (2003)

    Article  MathSciNet  Google Scholar 

  164. Pouzet, M., Sobrani, M.: Ordinal invariant of an age. in preparation

  165. Pouzet, M., Thiéry, N.: Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile. Electron. J. Combin. 20(2), Paper 1, 35 (2013)

  166. Pouzet, M., Thiéry, N.: Some relational structures with polynomial growth and their associated algebras II. Finite generation. Contrib. Discrete Math. 17, 70–108 (2022)

    Article  MathSciNet  Google Scholar 

  167. Pouzet, M., Zaguia, N.: Dimension de Krull des ensembles ordonnés. Special volume on ordered sets and their applications (L’Arbresle, 1982). Discrete Math. 53, 173–192 (1985)

  168. Pouzet, M., Zaguia, I.: On minimal prime graphs and posets. Order 26, 357–375 (2009)

    Article  MathSciNet  Google Scholar 

  169. Pouzet, M., Zaguia, I.: Graphs containing finite induced paths of unbounded length. DMTCS special volume edited by Karell Bertet, Miguel Couceiro, Jens Gustedt, Erkko Lehtonen, Nicolas Thiéry. 28pp. Vol 23:2, 3, (2021) arXiv:2011.00352v1

  170. Pouzet, M., Zaguia, I.: Hereditary classes of ordered sets coverable by two chains. 39 (2021). arXiv:2112.02633

  171. Priestley, H.A.: Representation of distributive lattices by means of ordered stone spaces. Bull. Lond. Math. Soc. 2, 507–530 (1970)

    Article  MathSciNet  Google Scholar 

  172. Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 24(3), 186–190 (1972)

    MathSciNet  Google Scholar 

  173. Priestley, H.A.: Ordered sets and duality for distributive lattices. Ann. Discrete Math. 23, 39–60 (1984)

    MathSciNet  Google Scholar 

  174. Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930)

    Article  MathSciNet  Google Scholar 

  175. Rathjen, M., Weiermann, A.: Proof-theoretic investigations on Kruskal’s theorem. Ann. Pure Appl. Logic 60, 49–88 (1993)

    Article  MathSciNet  Google Scholar 

  176. Robertson, N., Seymour, P.D.: Graph minors. A survey, Surveys in combinatorics 1985 (Glasgow, 1985), 153–171, London Math. Soc. Lecture Note Ser., 103, Cambridge Univ. Press, Cambridge, (1985)

  177. Robinson, A.: On the metamathematics of algebra, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, ix+195 (1951)

  178. Robinson, A.: Forcing in model theory, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 245–250. Gauthier-Villars, Paris, (1971)

  179. Rosenstein, J.G.: Linear orderings, Pure and Applied Mathematics, 98. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, xvii+487 (1982)

  180. Ryll-Nardzewski, C.: On categoricity in power \(\le \aleph _{\alpha }\). Bull. Acad. Pol. Sci. Sér. Math. Astr. Phys. 7, 545–548 (1959)

    MathSciNet  Google Scholar 

  181. Saracino, D.: Model companions for \(\aleph _0\)-categorical theories. Proc. Amer. Math. Soc. 39, 591–598 (1973)

    MathSciNet  Google Scholar 

  182. Scheinerman, E.R., Zito, J.: On the size of hereditary classes of graphs. J. Comb. Theory Ser. B 61, 16–39 (1994)

    Article  MathSciNet  Google Scholar 

  183. Schmerl, J.: Coinductive \(\aleph _0\)-categorical theories. J. Symb. Log. 55, 1130–1137 (1990)

    Article  MathSciNet  Google Scholar 

  184. Schmerl, J.H., Trotter, W.T.: Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures. Discrete Math. 113, 191–205 (1993)

    Article  MathSciNet  Google Scholar 

  185. Schmidt, D.: Well-partial orderings and their maximal order types. University of Heidelberg, Heidelberg, Habilitation (1978)

    Google Scholar 

  186. Sierpinski, W.: Sur les types d’ordre des ensembles linéaires. Fund. Math. 37, 253–264 (1950)

    Article  MathSciNet  Google Scholar 

  187. Slater, M.: On a class of order-types generalizing ordinals. Fund. Math. 54, 259–277 (1964)

    Article  MathSciNet  Google Scholar 

  188. Spielman, D., Bóna, M.: An infinite antichain of permutations. Electron. J. Combin. 7, Note 2, 4 (2000)

  189. Sobrani, M.: Structure d’ordre de la collection des âges de relations. Thèse de doctorat, Université Claude-Bernard, Lyon, 18 déc. (1992)

  190. Sobrani, M.: Sur les âges de relations et quelques aspects homologiques des constructions D+M. Thèse de doctorat d’état, Université S.M. Ben Abdallah-Fez, Fez, (2002)

  191. Stanley, R.P.: Enumerative combinatorics, vol. 1. Wadsworth - Brooks/Cole Advanced Books and Software, Belmont, Calif (1986)

    Book  Google Scholar 

  192. Sumner, D.P.: Graphs indecomposable with respect to the \(X\)-join. Discrete Math. 6, 281–298 (1973)

    Article  MathSciNet  Google Scholar 

  193. Tarski, A.: Contributions to the theory of models. III. Nederl. Akad. Wetensch. Proc. Ser. A. 58 (Indagationes Math. 17), 56–64 (1955)

  194. Tateno, A.: Mutually embeddable trees and a counterexample to the tree alternative conjecture, preprint, 32 (2008)

  195. Thomas, W.: Languages, automata, and logic, Handbook of formal languages, Vol. 3, 389–455, Springer, Berlin, (1997)

  196. Thomassé, S.: On better-quasi-ordering countable series-parallel orders. Trans. Amer. Math Soc. 352, 2491–2505 (1999)

    Article  MathSciNet  Google Scholar 

  197. Thomassé, S.: Conjectures on countable relations. preprint, 2000, 17 pp., and personal communication, November (2012)

  198. Trotter, W.T., Moore, J.I., Jr.: Some theorems on graphs and posets. Discrete Math. 15, 79–84 (1976)

    Article  MathSciNet  Google Scholar 

  199. Tyomkyn, M.: A proof of the rooted tree alternative conjecture. Discrete Math. 309, 5963–5967 (2009)

    Article  MathSciNet  Google Scholar 

  200. Vatter, V.: Permutation classes. In: Handbook of Enumerative Combinatorics , 754–833. CRC Press, (2015)

  201. Vaught, R.L.: Remarks on universal classes of relational systems. Nederl. Akad. Wetensch. Proc. Ser. A. 57 (Indagationes Math. 16), 589–591 (1954)

  202. Wolk, E.S.: Partially well-ordered sets and partial ordinals. Fund. Math. 60, 175–185 (1967)

    Article  MathSciNet  Google Scholar 

  203. Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods 3, 351–358 (1982)

    Article  MathSciNet  Google Scholar 

  204. Zaguia, N.: Chaînes d’idéaux et de sections initiales d’un ensemble ordonné. Publ. Dép. Math. (Lyon) (N.S.), (7/D) 97 (1983)

  205. Zverovich, I.: Extension of hereditary classes with substitutions. Discrete Appl. Math. 128, 487–509 (2003)

    Article  MathSciNet  Google Scholar 

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  1. ICJ, Mathématiques, Université Claude-Bernard Lyon1, 43 bd. 11 Novembre 1918, Lyon, Villeurbanne Cedex, 69622, France

    Maurice Pouzet

  2. Mathematics & Statistics Department, University of Calgary, Calgary, Alberta, T2N 1N4, Canada

    Maurice Pouzet

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$$\dagger $$ † To the memory of Roland Fraïssé.

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Pouzet, M. Well-quasi-ordering and Embeddability of Relational Structures. Order 41, 183–278 (2024). https://doi.org/10.1007/s11083-024-09664-y

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