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  • Print publication year: 2017
  • Online publication date: July 2017

6 - Ramsey-type and amalgamation-type properties of permutations



This survey deals with some aspects of combinatorics of permutations which are inspired by notions from structural Ramsey theory. Its first main focus is the overview of known results on Ramsey-type and Fraïssé-type properties of hereditary permutation classes, with particular emphasis on the concept of splittability. Secondly, we look at known estimates for Ramsey numbers of permutation matrices, and their relationship to Ramsey numbers of ordered graphs.


About this survey

Combinatorics of permutations is an old and well-established field of discrete mathematics. So is Ramsey theory. For a long time these two fields have followed their own separate ways without affecting each other much. However, in the first decade of this century, the situation started to slowly change, as concepts originating from the research on relational structures and on Ramsey classes became adopted (or, occasionally, reinvented) in the study of hereditary permutation classes.

The purpose of this paper is to give an introductory overview of the Ramsey-theoretic and relation-theoretic aspects of combinatorics of permutations, with particular focus on hereditary permutation classes. I do not assume any familiarity with either structural Ramsey theory or permutation combinatorics. In the rest of this first chapter, the reader will find a condensed introduction to the relevant notions from these fields.

Chapter 2 will then present a survey of the known results related to amalgamation, Ramseyness and other related properties of hereditary permutation classes.

The remaining two chapters contain a more detailed treatment of two specific topics related to amalgamation and Ramsey properties of permutations.

Chapter 3 focuses on the notion of unsplittability, a weak form of Ramsey property that has recently found applications in enumerative combinatorics of permutations, and appears to be a promising research direction.

Consequently, splittability and unsplittability play a prominent role in this survey; indeed, Chapter 3 is the longest of the four chapters.

Chapter 4 deals with estimates on Ramsey numbers of permutations. This topic, which is closely connected to graph theory, has received interest only very recently, with only a few nontrivial results known, and many basic questions still open.

[1] A. A., Ageev. A triangle-free circle graph with chromatic number 5. Discrete Math., 152(1–3):295–298, 1996.
[2] M. H., Albert. On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation. Random Structures Algorithms, 31(2):227–238, 2007.
[3] M. H., Albert, M. D., Atkinson, and M., Klazar. The enumeration of simple permutations. J. Integer Seq., 6, 2003. Article 03.4.4, 18 pages.
[4] M. H., Albert and V., Jelinek. Unsplittable classes of separable permutations. Electron. J. Combin., 23(2):#P2.49, 2016.
[5] M. H., Albert, J., Pantone, and V., Vatter. On the growth of merges and staircases of permutation classes. arXiv:1608.06969, 2016.
[6] M. D., Atkinson, M. M., Murphy, and N., Ruškuc. Partially well-ordered closed sets of permutations. Order, 19(2):101–113, 2002.
[7] M. D., Atkinson, M. M., Murphy, and N., Ruškuc. Pattern avoidance classes and subpermutations. Electron. J. Combin., 12:#R60, 2005.
[8] M., Balko, J., Cibulka, K., Kral, and J., Kynčl. Ramsey numbers of ordered graphs. Electron. Notes Discrete Math., 49:419–424, 2015.
[9] M., Balko, V., Jelinek, and P., Valtr. On ordered Ramsey numbers of bounded-degree graphs. arXiv:1606.05628, 2016.
[10] D., Bevan. Permutations avoiding 1324 and patterns in Łukasiewicz paths. J. Lond. Math. Soc., 92(1):105–122, 2015.
[11] M., Bodirsky. Ramsey classes: Examples and constructions. Surveys in Combinatorics 2015, (424):1, 2015.
[12] M., Bona. Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps. J. Combin. Theory Ser. A, 80(2):257–272, 1997.
[13] M., Bona. The limit of a Stanley–Wilf sequence is not always rational, and layered patterns beat monotone patterns. J. Combin. Theory Ser. A, 110(2):223–235, 2005.
[14] M., Bona. New records in Stanley–Wilf limits. European J. Combin., 28(1):75–85, 2007.
[15] M., Bona. A new upper bound for 1324-avoiding permutations. Combin. Probab. Comput., 23(5):717–724, 2014.
[16] M., Bona. A new record for 1324-avoiding permutations. Eur. J. Math., 1(1):198–206, 2015.
[17] J., Bottcher and J., Foniok. Ramsey properties of permutations. Electron. J. Combin., 20(1):#P2, 2013.
[18] S. A., Burr and P., Erdős. On the magnitude of generalized Ramsey numbers for graphs. In Infinite and Finite Sets, Vol. 1 (Keszthely 1973), volume 10 of Colloq. Math. Soc. Janos Bolyai, pages 214–240. North-Holland, Amsterdam, 1975.
[19] P. J., Cameron. Homogeneous permutations. Electron. J. Combin., 9(2):#R2, 2002.
[20] G. L., Cherlin. The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, volume 621 of Memoirs of the American Mathematical Society. American Mathematical Soc., 1998.
[21] V., Chvatal, V., Rodl, E., Szemeredi, and W. T., Trotter. The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B, 34(3):239–243, 1983.
[22] A., Claesson, V., Jelinek, and E., Steingrimsson. Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns. J. Combin. Theory Ser. A, 119:1680–1691, 2012.
[23] D., Conlon, J., Fox, C., Lee, and B., Sudakov. Ordered Ramsey numbers. J. Combin. Theory Ser. B, 122:353–383, 2017.
[24] D., Conlon, J., Fox, and B., Sudakov. Recent developments in graph Ramsey theory. Surveys in Combinatorics 2015, 424:49–118, 2015.
[25] J., Černy. Coloring circle graphs. Electron. Notes Discrete Math., 29(0):457–461, 2007.
[26] M., El-Zahar and N. W., Sauer. Ramsey-type properties of relational structures. Discrete Math., 94(1):1 – 10, 1991.
[27] J., Folkman. Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math., 18(1):19–24, 1970.
[28] J.|Fox. Stanley–Wilf limits are typically exponential. arXiv:1310.8378, 2013.
[29] R., Fraisse. Sur l'extension aux relations de quelques proprietes des ordres. Ann. Sci. Ec. Norm. Super., 71(4):363–388, 1954.
[30] I., Gessel. Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A, 53(2):257–285, 1990.
[31] R. L., Graham, B. L., Rothschild, and J. H., Spencer. Ramsey theory. John Wiley & Sons, 1990.
[32] S., Guillemot and D., Marx. Finding small patterns in permutations in linear time. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 82–101. ACM, New York, 2014.
[33] A., Gyarfas. On Ramsey covering numbers. In Infinite and Finite Sets (Keszthely 1973), volume 10 of Colloq. Math. Soc. Janos Bolyai, pages 801–816. North-Holland, Amsterdam, 1975.
[34] A., Gyarfas. On the chromatic number of multiple interval graphs and overlap graphs. Discrete Math., 55(2):161–166, 1985.
[35] A., Gyarfas. Corrigendum. Discrete Math., 62(3):333, 1986.
[36] S., Huczynska and N., Ruškuc. Pattern classes of permutations via bijections between linearly ordered sets. European J. Combin., 29(1):118–139, 2008.
[37] V., Jelinek and M., Klazar. Embedding dualities for set partitions and for relational structures. European J. Combin., 32(7):1084–1096, 2011.
[38] V., Jelinek and P., Valtr. Splittings and Ramsey properties of permutation classes. arXiv:1307.0027, 2013.
[39] V., Jelinek and P., Valtr. Splittings and Ramsey properties of permutation classes. Adv. Appl. Math., 63:41–67, 2015.
[40] A. S., Kechris, V. G., Pestov, and S., Todorcevic. Fraisse limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal., 15(1):106–189, 2005.
[41] H. A., Kierstead. Classes of graphs that are not vertex Ramsey. SIAM J. Discrete Math., 10(3):373–380, 1997.
[42] H. A., Kierstead and Y., Zhu. Classes of graphs that exclude a tree and a clique and are not vertex Ramsey. Combinatorica, 16(4):493–504, 1996.
[43] H. A., Kierstead and Y., Zhu. Radius three trees in graphs with large chromatic number. SIAM J. Discrete Math., 17(4):571–581, 2004.
[44] S., Kitaev and V., Lozin. Words and graphs. Springer, 2015.
[45] D. E., Knuth. The Art of Computer Programming, Volume 1 (3rd Ed.): Fundamental Algorithms. Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, USA, 1997.
[46] A., Kostochka. Upper bounds on the chromatic number of graphs. Trudy Instituta Matematiki (Novosibirsk), 10:204–226, 1988. (In Russian).
[47] A., Kostochka. Coloring intersection graphs of geometric figures with a given clique number. In Janos, Pach, editor, Towards a Theory of Geometric Graphs, volume 342 of Contemporary Mathematics, pages 127–138. Amer. Math. Soc., 2004.
[48] A., Kostochka and J., Kratochvil. Covering and coloring polygon-circle graphs. Discrete Math., 163(1–3):299–305, 1997.
[49] A. H., Lachlan and R. E., Woodrow. Countable ultrahomogeneous undirected graphs. Trans. Amer. Math. Soc., pages 51–94, 1980.
[50] C., Lee. Ramsey numbers of degenerate graphs. arXiv:1505.04773, 2015. To appear in Annals of Mathematics.
[51] D., Macpherson. A survey of homogeneous structures. Discrete Math., 311(15):1599–1634, 2011.
[52] A., Marcus and G., Tardos. Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004.
[53] M. M., Murphy. Restricted permutations, antichains, atomic classes, and stack sorting. PhD thesis, University of St. Andrews, 2002.
[54] G. V., Nenashev. An upper bound on the chromatic number of a circle graph without K4. J. Math. Sci. (N.Y.), 184(5):629–633, 2012.
[55] J., Nešetřil and V., Rodl. Partitions of vertices. Commentationes Mathematicae Universitatis Carolinae, 17(1):85–95, 1976.
[56] J., Nešetřil and V., Rodl. On Ramsey graphs without cycles of short odd lengths. Commentationes Mathematicae Universitatis Carolinae, 20(3):565–582, 1979.
[57] J., Nešetřil and V., Rodl. Simple proof of the existence of restricted Ramsey graphs by means of a partite construction. Combinatorica, 1(2):199–202, 1981.
[58] J., Nešetřil. Ramsey classes and homogeneous structures. Combin. Probab. Comput., 14(1):171–189, January 2005.
[59] M., Opler. Personal communication, 2016.
[60] J., Pach and G., Tardos. Forbidden paths and cycles in ordered graphs and matrices. Israel J. Math., 155(1):359–380, 2006.
[61] F. P., Ramsey. On a problem of formal logic. Proc. Lond. Math. Soc., s2-30(1):264–286, 1930.
[62] V., Rodl and N., Sauer. The Ramsey property for families of graphs which exclude a given graph. Canad. J. Math., 44:1050–1060, 1992.
[63] V., Rodl, N., Sauer, and X., Zhu. Ramsey families which exclude a graph. Combinatorica, 15(4):589–596, 1995.
[64] N., Sauer. Vertex partition problems. In D., Miklos, V. T., Sos, and T., Szőnyi, editors, Combinatorics, Paul Erdős is eighty, Vol. 1, pages 361–377. Janos Bolyai Mathematical Society, 1993.
[65] N., Sauer. On the Ramsey property of families of graphs. Trans. Amer. Math. Soc., 347(3):785–833, 1995.
[66] N., Sauer. Age and weak indivisibility. European J. Combin., 37:24–31, 2014.
[67] J. H., Schmerl. Countable homogeneous partially ordered sets. Algebra Universalis, 9(1):317–321, 1979.
[68] M., Sokić. Ramsey Property of Posets and Related Structures. PhD thesis, University of Toronto, 2010.
[69] M., Sokić. Ramsey property, ultrametric spaces, finite posets, and universal minimal flows. Israel J. Math., 194(2):609–640, 2013.
[70] S., Solecki and M., Zhao. A Ramsey theorem for partial orders with linear extensions. European J. Combin., 60:21–30, 2017.
[71] D. P., Sumner. Subtrees of a graph and chromatic number. In G., Chartrand, Y., Alavi, D. L., Goldsmith, L., Lesniak-Foster, and D. L., Lick, editors, The theory and applications of graphs (Kalamazoo, MI, 1980), pages 557–576. John Wiley and Sons Dordrecht and Boston, 1981.
[72] V., Vatter. Small permutation classes. Proc. Lond. Math. Soc., 103(5):879–921, 2011.
[73] V., Vatter. Permutation classes. In Handbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), page 753833. CRC Press, Boca Raton, FL, 2015.
[74] V., Vatter. An Erdős–Hajnal analogue for permutation classes. Discrete Math. Theor. Comput. Sci., 8(2):#4, 2016.
[75] A., Zucker. Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem. Trans. Amer. Math. Soc., 368(9):6715–6740, 2016.