Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-05T04:27:02.205Z Has data issue: false hasContentIssue false
  • English
  • Français

A Short Proof of the Random Ramsey Theorem

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  22 December 2014

RAJKO NENADOV  and
ANGELIKA STEGER
RAJKO NENADOV
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail: [email protected], [email protected])
ANGELIKA STEGER
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that

$$ \begin{equation*} \Pr[G_{n,p} \rightarrow (F)_r^e] = \begin{cases} 1-o(1) &p\ge Cn^{-1/m_2(F)},\\ o(1) &p\le cn^{-1/m_2(F)}, \end{cases} \end{equation*} $$
where
$$ \begin{equation*} m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}. \end{equation*} $$
 The proof of the 1-statement is based on the recent beautiful hypergraph container theorems by Saxton and Thomason, and Balogh, Morris and Samotij. The proof of the 0-statement is elementary.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 130 - 144
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

[1]Balogh, J., Morris, R. and Samotij, W. (2012) Independent sets in hypergraphs. J. Amer. Math. Soc. arXiv:1204.6530, to appear.Google Scholar
[2]Chen, B., Matsumoto, M., Wang, J. F., Zhang, Z. F. and Zhang, J. X. (1994) A short proof of Nash-Williams' theorem for the arboricity of a graph. Graphs Combin. 10 27–28.CrossRefGoogle Scholar
[3]Conlon, D. and Gowers, T. Combinatorial theorems in sparse random sets. arXiv:1011.4310Google Scholar
[4]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
[5]Friedgut, E., Rödl, V. and Schacht, M. (2010) Ramsey properties of random discrete structures. Random Struct. Alg. 37 407–436.CrossRefGoogle Scholar
[6]Gugelmann, L., Person, Y., Steger, A. and Thomas, H. (2012) A randomized version of Ramsey's theorem. Random Struct. Alg. 41 488–505.CrossRefGoogle Scholar
[7]Łuczak, T., Ruciński, A. and Voigt, B. (1992) Ramsey properties of random graphs. J. Combin. Theory Ser. B 56 55–68.CrossRefGoogle Scholar
[8]Marciniszyn, M., Skokan, J., Spöhel, R. and Steger, A. (2009) Asymmetric Ramsey properties of random graphs involving cliques. Random Struct. Alg. 34 419–453.CrossRefGoogle Scholar
[9]Rödl, V. and Ruciński, A. (1993) Lower bounds on probability thresholds for Ramsey properties. In Combinatorics: Paul Erdős is Eighty, Vol. 1, Bolyai Society Mathematical Studies, pp. 317–346.Google Scholar
[10]Rödl, V. and Ruciński, A. (1994) Random graphs with monochromatic triangles in every edge coloring. Random Struct. Alg. 5 253–270.CrossRefGoogle Scholar
[11]Rödl, V. and Ruciński, A. (1995) Threshold functions for Ramsey properties. J. Amer. Math. Soc. 8 917–942.CrossRefGoogle Scholar
[12]Saxton, D. and Thomason, A. (2012) Hypergraph containers. arXiv:1204.6595Google Scholar