# Turán-Ramsey Theorems and Kp-Independence Numbers

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

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8 Citations (Scopus)

### Abstract

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  and , 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 language English 297-325 29 Combinatorics Probability and Computing 3 3 https://doi.org/10.1017/S0963548300001218 Published - 1994

### ASJC Scopus subject areas

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