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Ramsey numbers of ordered graphs. (English) Zbl 1440.05146

Summary: An ordered graph is a pair \(\mathcal{G}=(G,\prec)\) where \(G\) is a graph and \(\prec\) is a total ordering of its vertices. The ordered Ramsey number \(\overline{R}(\mathcal{G})\) is the minimum number \(N\) such that every ordered complete graph with \(N\) vertices and with edges colored by two colors contains a monochromatic copy of \(\mathcal{G} \).
In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings \(\mathcal{M}_n\) on \(n\) vertices for which \(\overline{R}(\mathcal{M}_n)\) is superpolynomial in \(n\). This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number \(\overline{R}(\mathcal{G})\) is polynomial in the number of vertices of \(\mathcal{G}\) if the bandwidth of \(\mathcal{G}\) is constant or if \(\mathcal{G}\) is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of D. Conlon et al. [J. Comb. Theory, Ser. B 122, 353–383 (2017; Zbl 1350.05085)].
For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by G. Károlyi et al. [Discrete Comput. Geom. 20, No. 3, 375–388 (1998; Zbl 0912.05046)].

MSC:

05C55 Generalized Ramsey theory
05D10 Ramsey theory

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[1] G. Audemard and L. Simon, Glucose 2.3 in the SAT 2013 Competition,Department of Computer Science Series of Publications B vol.B-2013-1, University of Helsinki (2013), 42-43.
[2] M. Balko, J. Cibulka, K. Kr´al, and J. Kynˇcl, Ramsey numbers of ordered graphs, Electron. Notes Discrete Math.49(2015), 419-424. · Zbl 1346.05179
[3] M. Balko, J. Cibulka, K. Kr´al, and J. Kynˇcl,http://kam.mff.cuni.cz/ balko/ ordered_ramsey
[4] B. Bollob´as,Extremal Graph Theory, New York: Dover publications, ISBN 978-0486-43596-1 (2004). · Zbl 1099.05044
[5] J. A. Bondy and P. Erd˝os, Ramsey numbers for cycles in graphs,J. Combin. Theory Ser. B14(1) (1973), 46-54. · Zbl 0248.05127
[6] S. A. Burr and J. A. Roberts, On Ramsey numbers for stars,Utilitas Math.4(1973), 217-220. · Zbl 0293.05119
[7] G. Chartrand and S. Schuster, On the existence of specified cycles in complementary graphs,Bull. Amer. Math. Soc.77(1971), 995-998. · Zbl 0224.05121
[8] S. A. Choudum and B. Ponnusamy, Ordered Ramsey numbers,Discrete Math.247(1- 3) (2002), 79-92. · Zbl 0998.05046
[9] F. R. K. Chung, F. T. Leighton, and A. L. Rosenberg, Embedding graphs in books: a layout problem with applications to VLSI design,SIAM J. on Algebraic Discrete Methods8(1) (1987), 33-58. · Zbl 0617.68062
[10] V. Chv´atal and F. Harary, Generalized Ramsey theory for graphs, II, Small diagonal numbers,Proc. Amer. Math. Soc.32(2) (1972), 389-394. · Zbl 0229.05116
[11] V. Chv´atal, V. R¨odl, E. Szemer´edi, and W. T. Trotter Jr., The Ramsey number of a graph with bounded maximum degree,J. Combin. Theory Ser. B34(3) (1983), 239-243. · Zbl 0547.05044
[12] J. Cibulka, Extremal combinatorics of matrices, sequences and sets of permutations, Ph.D. Thesis, Charles University, Prague (2013). · Zbl 1268.05225
[13] J. Cibulka, P. Gao, M. Krˇc´al, T. Valla, and P. Valtr, On the geometric Ramsey number of outerplanar graphs,Discrete Comput. Geom.53(1) (2015), 64-79. · Zbl 1309.05124
[14] D. Conlon, A new upper bound for diagonal Ramsey numbers,Ann. of Math.170(2) (2009), 941-960. · Zbl 1188.05087
[15] D. Conlon, J. Fox, C. Lee, and B. Sudakov, Ordered Ramsey numbers,J. Combin. Theory Ser. B122(2017), 353-383. · Zbl 1350.05085
[16] D. Conlon, J. Fox, and B. Sudakov, Ramsey numbers of sparse hypergraphs,Random Structures Algorithms35(1) (2009), 1-14. · Zbl 1203.05099
[17] D. Conlon, J. Fox, and B. Sudakov, Recent developments in graph Ramsey theory, Surveys in combinatorics 2015, 49-118, London Math. Soc. Lecture Note Ser., 424, Cambridge Univ. Press, Cambridge, 2015. · Zbl 1352.05123
[18] M. Eli´aˇs and J. Matouˇsek, Higher-order Erd˝os-Szekeres theorems,Adv. Math.244 (2013), 1-15. · Zbl 1283.05175
[19] P. Erd˝os, Some remarks on the theory of graphs,Bull. Amer. Math. Soc53(4) (1947), 292-294. · Zbl 0032.19203
[20] P. Erd˝os and G. Szekeres, A combinatorial problem in geometry,Compos. Math2 (1935), 463-470.
[21] R. J. Faudree and R. H. Schelp, All Ramsey numbers for cycles in graphs,Discrete Math.8(4) (1974), 313-329. · Zbl 0294.05122
[22] J. Fox, J. Pach, B. Sudakov, and A. Suk, Erd˝os-Szekeres-type theorems for monotone paths and convex bodies,Proc. London Math. Soc.105(5) (2012), 953-982. · Zbl 1254.05114
[23] Z. F¨uredi and P. Hajnal, Davenport-Schinzel theory of matrices,Discrete Math. 103(3) (1992), 233-251. · Zbl 0776.05024
[24] L. Gerencs´er and L. Gy´arf´as, On Ramsey-type problems,Ann. Univ. Sci. E¨otv¨os Sect. Math.10(1967), 167-170. · Zbl 0163.45502
[25] R. Graham and J. Neˇsetˇril, Ramsey theory and Paul Erd˝os (recent results from a historical perspective),Paul Erd˝os and his mathematics. II (Budapest, 1999), 339- 365,Bolyai Soc. Math. Stud.11, J´anos Bolyai Math. Soc., Budapest (2002). · Zbl 1037.05047
[26] C. Hylt´en-Cavallius, On a combinatorial problem,Colloq. Math.6(1958), 59-65. · Zbl 0086.01202
[27] Gy. K´arolyi, J. Pach, and G. T´oth, Ramsey-type results for geometric graphs, I, Discrete Comput. Geom.18(3) (1997), 247-255. · Zbl 0940.05046
[28] Gy. K´arolyi, J. Pach, G. T´oth, and P. Valtr, Ramsey-type results for geometric graphs, II,Discrete Comput. Geom.20(3) (1998), 375-388. · Zbl 0912.05046
[29] M. Klazar, Extremal problems for ordered (hyper)graphs: applications of Davenport- Schinzel sequences,European J. Combin.25(1) (2004), 125-140. · Zbl 1031.05071
[30] M. Klazar, Extremal problems for ordered hypergraphs: small patterns and some enumeration,Discrete Appl. Math.143(1-3) (2004), 144-154. · Zbl 1054.05072
[31] T. K˝ov´ari, V. S´os, and P. Tur´an, On a problem of K. Zarankiewicz,Colloq. Math.,3 (1954), 50-57. · Zbl 0055.00704
[32] K. G. Milans, D. Stolee, and D. B. West, Ordered Ramsey theory and track representations of graphs,J. Comb.,6(4) (2015), 445-456. · Zbl 1325.05111
[33] G. Moshkovitz and A. Shapira, Ramsey theory, integer partitions and a new proof of the Erd˝os-Szekeres theorem,Adv. in Math.262(2014), 1107-1129. · Zbl 1295.05255
[34] J. Pach and G. Tardos, Forbidden paths and cycles in ordered graphs and matrices, Israel J. Math.155(1) (2006), 359-380. · Zbl 1132.05035
[35] F. P. Ramsey, On a Problem of Formal Logic,Proc. London Math. Soc.S2-30(1) (1930), 264-286.
[36] V. Rosta, On a Ramsey-type problem of J. A. Bondy and P. Erd˝os,J. Combin. Theory Ser. B15(1) (1973), 94-120. · Zbl 0261.05119
[37] J.
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