Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2025-01-05T12:00:47.449Z Has data issue: false hasContentIssue false
  • English
  • Français

Monochromatic trees in random tournaments

Part of: Graph theory

Published online by Cambridge University Press:  07 November 2019

Matija Bucić ,
Sven Heberle ,
Shoham Letzter  and
Benny Sudakov
Matija Bucić
Affiliation:
Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland
Sven Heberle
Affiliation:
Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland
Shoham Letzter
Affiliation:
ETH Institute for Theoretical Studies, ETH Zürich, Clausiusstrasse 47, 8092 Zürich, Switzerland
Benny Sudakov*
Affiliation:
Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland
*
*Corresponding author. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.

MSC classification

Primary: 05C05: Trees
Secondary: 05C55: Generalized Ramsey theory 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 3 , May 2020 , pp. 318 - 345
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Research supported by Dr Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation.

‡

Research supported in part by SNSF grant 200021-175573.

References

Alon, N. and Shapira, A. (2004) Testing subgraphs in directed graphs. J. Comput. Syst. Sci. 69 354–382.CrossRefGoogle Scholar
Ben-Eliezer, I., Krivelevich, M. and Sudakov, B. (2012) The size Ramsey number of a directed path. J. Combin. Theory Ser. B 102 743–755.CrossRefGoogle Scholar
Bermond, J.-C. (1974) Some Ramsey numbers for directed graphs. Discrete Math . 9 313–321.CrossRefGoogle Scholar
Bucić, M., Letzter, S. and Sudakov, B. (2019) Directed Ramsey number for trees. J. Combin. Theory Ser. B 137 145–177.CrossRefGoogle Scholar
Bucić, M., Letzter, S. and Sudakov, B. (2019) Monochromatic paths in random tournaments. Random Struct. Alg . 54 69–81.CrossRefGoogle Scholar
Burr, S. A. and Erdös, P. (1975) On the magnitude of generalized Ramsey numbers for graphs. Colloq. Math. Soc. János Bolyai 10 215–240.Google Scholar
Chvatál, V., Rödl, V., Szemerédi, E. and Trotter, W. T. (1983) The Ramsey number of a graph with bounded maximum degree. J. Combin. Theory Ser. B 34 239–243.CrossRefGoogle Scholar
Conlon, D., Fox, J. and Sudakov, B. (2012) On two problems in graph Ramsey theory. Combinatorica 32 513–535.CrossRefGoogle Scholar
Eaton, N. (1998) Ramsey numbers for sparse graphs. Discrete Math . 185 63–75.CrossRefGoogle Scholar
Gallai, T. (1968) On directed paths and circuits. In Theory of Graphs (Proc. Colloq., Tihany, 1966) (P. Erdös and G. Katona, eds), Academic Press, pp. 115–118.Google Scholar
Graham, R. L., Rödl, V. and Ruciński, A. (2000) On graphs with linear Ramsey numbers. J. Graph Theory 35 176–192.3.0.CO;2-C>CrossRefGoogle Scholar
Gyárfás, A. and Lehel, J. (1973) A Ramsey-type problem in directed and bipartite graphs. Period. Math. Hungar. 3 299–304.CrossRefGoogle Scholar
Harary, F. and Hell, P. (1974) Generalized Ramsey theory for graphs, V: The Ramsey number of a digraph. Bull. London Math. Soc. 6 175–182.CrossRefGoogle Scholar
Hasse, M. (1965) Zur algebraischen Begründung der Graphentheorie, I. Math. Nachr. 28 275–290.CrossRefGoogle Scholar
Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc. 30 264–285.Google Scholar
Raynaud, H. (1973) Sur le circuit hamiltonien bi-coloré dans les graphes orientés. Period. Math. Hungar. 3 289–297.CrossRefGoogle Scholar
Roy, B. (1967) Nombre chromatique et plus longs chemins d’un graphe. Rev. Française Informat. Recherche Opérationnelle 1 129–132.Google Scholar
Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes, Vol. 260 of Proc. Colloq. Internat. CNRS, pp. 399–401.Google Scholar
Vitaver, L. M. (1962) Determination of minimal coloring of vertices of a graph by means of Boolean powers of the incidence matrix. Dokl. Akad. Nauk SSSR 147 758–759.Google Scholar
Williamson, J. E. (1973) A Ramsey type problem for paths in digraphs. Math. Ann. 203 117–118.CrossRefGoogle Scholar