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Large Unavoidable Subtournaments

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  21 June 2016

EOIN LONG
EOIN LONG*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: eoinlong@post.tau.ac.il)
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Abstract

Let Dk denote the tournament on 3k vertices consisting of three disjoint vertex classes V 1, V 2 and V 3 of size k, each oriented as a transitive subtournament, and with edges directed from V 1 to V 2, from V 2 to V 3 and from V 3 to V 1. Fox and Sudakov proved that given a natural number k and ε > 0, there is n 0(k, ε) such that every tournament of order n ⩾ n 0(k,ε) which is ε-far from being transitive contains Dk as a subtournament. Their proof showed that $n_0(k,\epsilon ) \leq \epsilon ^{-O(k/\epsilon ^2)}$ and they conjectured that this could be reduced to n 0(k, ε) ⩽ ε−O(k). Here we prove this conjecture.

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05C20: Directed graphs (digraphs), tournaments

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 1 , January 2017 , pp. 68 - 77
Copyright
Copyright © Cambridge University Press 2016 

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References

[1] Agarwal, P. and Pach, J. (1995) Combinatorial Geometry, Wiley.Google Scholar
[2] Alon, N. and Spencer, J. (2008) The Probabilistic Method, third edition, Wiley.CrossRefGoogle Scholar
[3] Berger, E., Choromanski, K., Chudnovksy, M., Fox, J., Loebl, M., Scott, A., Seymour, P. and Thomassé, S. (2013) Tournaments and colouring. J. Combin. Theory Ser. B 103 1–20.CrossRefGoogle Scholar
[4] Conlon, D. (2009) A new upper bound for diagonal Ramsey numbers. Ann. of Math. 170 941–960.CrossRefGoogle Scholar
[5] Cutler, J. and Montágh, B. (2008) Unavoidable subgraphs of colored graphs. Discrete Math. 308 4396–4413.Google Scholar
[6] Erdős, P. (1947) Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53 292–294.CrossRefGoogle Scholar
[7] Erdős, P. and Moser, L. (1964) On the representation of directed graphs as unions of orderings. Publ. Math. Inst. Hungar. Acad. Sci. 9 125–132.Google Scholar
[8] Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Mathematica 2 463–470.Google Scholar
[9] Fox, J. and Sudakov, B. (2008) Unavoidable patterns. J. Combin. Theory Ser. A 115 1561–1569.Google Scholar
[10] Fox, J. and Sudakov, B. (2011) Dependent random choice. Random Struct. Alg. 38 68–99.CrossRefGoogle Scholar
[11] Furstenberg, H. (1981) Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press.Google Scholar
[12] Kövari, T., Sós, V. T. and Turán, P. (1954) On a problem of K. Zarankiewicz. Colloq. Math. 3 50–57.CrossRefGoogle Scholar
[13] Graham, R. L., Rothschild, B. L. and Spencer, J. H. (1990) Ramsey Theory, second edition, Wiley.Google Scholar
[14] Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc. 30 264–286.Google Scholar
[15] Shapira, A. and Yuster, R. (2016) Unavoidable tournaments. J. Combin. Theory Ser. B, 116 191–207.Google Scholar
[16] Spencer, J. (1975) Ramsey's theorem: A new lower bound. J. Combin. Theory Ser. A 18 108–115.Google Scholar
[17] Zarankiewicz, K. (1951) Problem P 101. Colloq. Math. 2 301.Google Scholar