Turán-Ramsey Theorems and Kp-Independence Numbers

P. Erdős, A. Hajnal, M. Simonovits, V. T. Sós, E. Szemerédi

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Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices. The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value of [formula omitted] for p <q. Several results are proved, and some other problems and conjectures are stated.

Original languageEnglish
Pages (from-to)297-325
Number of pages29
JournalCombinatorics Probability and Computing
Issue number3
Publication statusPublished - 1994


ASJC Scopus subject areas

  • Applied Mathematics
  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics

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