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Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs

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Abstract

Letk t(G) be the number of cliques of ordert in the graphG. For a graphG withn vertices let\(c_t (G) = \frac{{k_t (G) + k_t (\bar G)}}{{\left( {\begin{array}{*{20}c} n \\ t \\ \end{array} } \right)}}\). Letc t(n)=Min{c t(G)∶∇G∇=n} and let\(c_t = \mathop {\lim }\limits_{n \to \infty } c_t (n)\). An old conjecture of Erdös [2], related to Ramsey's theorem states thatc t=21-(t/2). Recently it was shown to be false by A. Thomason [12]. It is known thatc t(G)≈21-(t/2) wheneverG is a pseudorandom graph. Pseudorandom graphs — the graphs “which behave like random graphs” — were inroduced and studied in [1] and [13]. The aim of this paper is to show that fort=4,c t(G)≥21-(t/2) ifG is a graph arising from pseudorandom by a small perturbation.

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Authors and Affiliations

  1. Department of Computer Science and Systems, McMaster University, L8S 4K1, Hamilton, Ontario, Canada

    F. Franek

  2. Departments of Mathematics and Computer Science Emory University, 30322, Atlanta, Georgia, USA

    V. Rödl

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  1. F. Franek
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  2. V. Rödl
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Research supported by NSERC grant OGP0025112

Research supported by NSF grant DMS 9011850

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Franek, F., Rödl, V. Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs. Graphs and Combinatorics 8, 299–308 (1992). https://doi.org/10.1007/BF02351585

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