Book contents
- Frontmatter
- Contents
- Preface
- 1 Relations among partitions
- 2 Large-scale structures in random graphs
- 3 The spt-function of Andrews
- 4 Combinatorial structures in finite classical polar spaces
- 5 Switching techniques for edge decompositions of graphs
- 6 Ramsey-type and amalgamation-type properties of permutations
- 7 Monotone cellular automata
- 8 Robustness of graph properties
- 9 Some applications of relative entropy in additive combinatorics
- References
6 - Ramsey-type and amalgamation-type properties of permutations
Published online by Cambridge University Press: 21 July 2017
By
Book contents
- Frontmatter
- Contents
- Preface
- 1 Relations among partitions
- 2 Large-scale structures in random graphs
- 3 The spt-function of Andrews
- 4 Combinatorial structures in finite classical polar spaces
- 5 Switching techniques for edge decompositions of graphs
- 6 Ramsey-type and amalgamation-type properties of permutations
- 7 Monotone cellular automata
- 8 Robustness of graph properties
- 9 Some applications of relative entropy in additive combinatorics
- References
Summary
Abstract
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.
Introduction
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.
- Type
- Chapter
- Information
- Surveys in Combinatorics 2017 , pp. 272 - 311Publisher: Cambridge University PressPrint publication year: 2017
References
[1] . A triangle-free circle graph with chromatic number 5. Discrete Math., 152(1–3):295–298, 1996.Google Scholar
[2] . On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation. Random Structures Algorithms, 31(2):227–238, 2007.Google Scholar
[3] , , and . The enumeration of simple permutations. J. Integer Seq., 6, 2003. Article 03.4.4, 18 pages.Google Scholar
[4] and . Unsplittable classes of separable permutations. Electron. J. Combin., 23(2):#P2.49, 2016.Google Scholar
[5] , , and . On the growth of merges and staircases of permutation classes. arXiv:1608.06969, 2016.
[6] , , and . Partially well-ordered closed sets of permutations.
Order, 19(2):101–113, 2002.Google Scholar
[7] , , and . Pattern avoidance classes and subpermutations. Electron. J. Combin., 12:#R60, 2005.Google Scholar
[8] , , , and . Ramsey numbers of ordered graphs. Electron. Notes Discrete Math., 49:419–424, 2015.Google Scholar
[9] , , and . On ordered Ramsey numbers of bounded-degree graphs. arXiv:1606.05628, 2016.
[10] . Permutations avoiding 1324 and patterns in Łukasiewicz paths. J. Lond. Math. Soc., 92(1):105–122, 2015.Google Scholar
[11] . Ramsey classes: Examples and constructions. Surveys in Combinatorics 2015, (424):1, 2015.Google Scholar
[12] . 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.Google Scholar
[13]
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.Google Scholar
[15] . A new upper bound for 1324-avoiding permutations. Combin. Probab. Comput., 23(5):717–724, 2014.Google Scholar
[16] . A new record for 1324-avoiding permutations. Eur. J. Math., 1(1):198–206, 2015.Google Scholar
[17] and . Ramsey properties of permutations. Electron. J. Combin., 20(1):#P2, 2013.
[18] and . 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.
[20] . 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] , , , and . The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B, 34(3):239–243, 1983.Google Scholar
[22] , , and . Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns. J. Combin. Theory Ser. A, 119:1680–1691, 2012.Google Scholar
[24] , , and . Recent developments in graph Ramsey theory. Surveys in Combinatorics 2015, 424:49–118, 2015.Google Scholar
[26] and . Ramsey-type properties of relational structures. Discrete Math., 94(1):1 – 10, 1991.Google Scholar
[27] . Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math., 18(1):19–24, 1970.Google Scholar
[28] J.|Fox. Stanley–Wilf limits are typically exponential. arXiv:1310.8378, 2013.
[29] . Sur l'extension aux relations de quelques proprietes des ordres.
Ann. Sci. Ec. Norm. Super., 71(4):363–388, 1954.Google Scholar
[30] . Symmetric functions and P-recursiveness.
J. Combin. Theory Ser. A, 53(2):257–285, 1990.Google Scholar
[31] , , and . Ramsey theory. John Wiley & Sons, 1990.
[32] and . 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] . 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] . On the chromatic number of multiple interval graphs and overlap graphs.
Discrete Math., 55(2):161–166, 1985.Google Scholar
[35] . Corrigendum. Discrete Math., 62(3):333, 1986.
[36] and . Pattern classes of permutations via bijections between linearly ordered sets. European J. Combin., 29(1):118–139, 2008.Google Scholar
[37] and . Embedding dualities for set partitions and for relational structures.
European J. Combin., 32(7):1084–1096, 2011.Google Scholar
[38] and . Splittings and Ramsey properties of permutation classes. arXiv:1307.0027, 2013.
[39] and . Splittings and Ramsey properties of permutation classes. Adv. Appl. Math., 63:41–67, 2015.Google Scholar
[40] , , and . Fraisse limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal., 15(1):106–189, 2005.Google Scholar
[41] . Classes of graphs that are not vertex Ramsey. SIAM J. Discrete Math., 10(3):373–380, 1997.Google Scholar
[42] and . Classes of graphs that exclude a tree and a clique and are not vertex Ramsey. Combinatorica, 16(4):493–504, 1996.Google Scholar
[43] and . Radius three trees in graphs with large chromatic number. SIAM J. Discrete Math., 17(4):571–581, 2004.Google Scholar
[44] and . Words and graphs. Springer, 2015.
[45] . The Art of Computer Programming, Volume 1 (3rd Ed.): Fundamental Algorithms. Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, USA, 1997.
[46] . Upper bounds on the chromatic number of graphs. Trudy Instituta Matematiki (Novosibirsk), 10:204–226, 1988. (In Russian).Google Scholar
[47] . Coloring intersection graphs of geometric figures with a given clique number. In , editor, Towards a Theory of Geometric Graphs, volume 342 of Contemporary Mathematics, pages 127–138. Amer. Math. Soc., 2004.
[48] and . Covering and coloring polygon-circle graphs. Discrete Math., 163(1–3):299–305, 1997.Google Scholar
[49] and . Countable ultrahomogeneous undirected graphs. Trans. Amer. Math. Soc., pages 51–94, 1980.Google Scholar
[50] . Ramsey numbers of degenerate graphs. arXiv:1505.04773, 2015. To appear in Annals of Mathematics.
[52] and . Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004.Google Scholar
[53] . Restricted permutations, antichains, atomic classes, and stack sorting. PhD thesis, University of St. Andrews, 2002.
[54] . An upper bound on the chromatic number of a circle graph without K4. J. Math. Sci. (N.Y.), 184(5):629–633, 2012.Google Scholar
[55] and . Partitions of vertices. Commentationes Mathematicae Universitatis Carolinae, 17(1):85–95, 1976.Google Scholar
[56] and . On Ramsey graphs without cycles of short odd lengths. Commentationes Mathematicae Universitatis Carolinae, 20(3):565–582, 1979.Google Scholar
[57] and . Simple proof of the existence of restricted Ramsey graphs by means of a partite construction. Combinatorica, 1(2):199–202, 1981.Google Scholar
[58] . Ramsey classes and homogeneous structures. Combin. Probab. Comput., 14(1):171–189, January 2005.Google Scholar
[59] . Personal communication, 2016.
[60] and . Forbidden paths and cycles in ordered graphs and matrices. Israel J. Math., 155(1):359–380, 2006.Google Scholar
[62] and . The Ramsey property for families of graphs which exclude a given graph. Canad. J. Math., 44:1050–1060, 1992.Google Scholar
[63] , , and . Ramsey families which exclude a graph. Combinatorica, 15(4):589–596, 1995.Google Scholar
[64] . Vertex partition problems. In , , and , editors, Combinatorics, Paul Erdős is eighty, Vol. 1, pages 361–377. Janos Bolyai Mathematical Society, 1993.
[65] . On the Ramsey property of families of graphs.
Trans. Amer. Math. Soc., 347(3):785–833, 1995.Google Scholar
[67] . Countable homogeneous partially ordered sets. Algebra Universalis, 9(1):317–321, 1979.Google Scholar
[68] . Ramsey Property of Posets and Related Structures. PhD thesis, University of Toronto, 2010.
[69] . Ramsey property, ultrametric spaces, finite posets, and universal minimal flows. Israel J. Math., 194(2):609–640, 2013.Google Scholar
[70] and . A Ramsey theorem for partial orders with linear extensions. European J. Combin., 60:21–30, 2017.Google Scholar
[71] . Subtrees of a graph and chromatic number. In , , , , and , editors, The theory and applications of graphs (Kalamazoo, MI, 1980), pages 557–576. John Wiley and Sons Dordrecht and Boston, 1981.
[73] . Permutation classes. In Handbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), page 753833. CRC Press, Boca Raton, FL, 2015.
[74] . An Erdős–Hajnal analogue for permutation classes. Discrete Math. Theor. Comput. Sci., 8(2):#4, 2016.Google Scholar
[75] . Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem.
Trans. Amer. Math. Soc., 368(9):6715–6740, 2016.Google Scholar