Turán-Ramsey theorems and simple asymptotically extremal structures

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

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

This paper is a continuation of [10], where P. Erdo{combining double acute accent}s, A. Hajnal, V. T. Sós, and E. Szemerédi investigated the following problem: Assume that a so called forbidden graph L and a function f(n)=o(n) are fixed. What is the maximum number of edges a graph Gn on n vertices can have without containing L as a subgraph, and also without having more than f(n) independent vertices? This problem is motivated by the classical Turán and Ramsey theorems, and also by some applications of the Turán theorem to geometry, analysis (in particular, potential theory) [27-29], [11-13]. In this paper we are primarily interested in the following problem. Let (Gn) be a graph sequence where Gn has n vertices and the edges of Gn are coloured by the colours χ1,...,χr so that the subgraph of colour χυ contains no complete subgraph Kpv, (v=1,... r). Further, assume that the size of any independent set in Gn is o(n) (as n→∞). What is the maximum number of edges in Gn under these conditions? One of the main results of this paper is the description of a procedure yielding relatively simple sequences of asymptotically extremal graphs for the problem. In a continuation of this paper we shall investigate the problem where instead of α(Gn)=o(n) we assume the stronger condition that the maximum size of a Kp-free induced subgraph of Gn is o(n).

Original languageEnglish
Pages (from-to)31-56
Number of pages26
JournalCombinatorica
Volume13
Issue number1
DOIs
Publication statusPublished - Mar 1993

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Ramsey's Theorem
Color
Subgraph
Continuation
Graph in graph theory
Geometry
Extremal Graphs
Potential Theory
Induced Subgraph
Independent Set
Acute
Theorem

Keywords

  • AMS subject classification code (1991): 05C35, 05C55

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Mathematics(all)

Cite this

Turán-Ramsey theorems and simple asymptotically extremal structures. / Erdős, P.; Hajnal, A.; Simonovits, M.; Sós, V. T.; Szemerédi, E.

In: Combinatorica, Vol. 13, No. 1, 03.1993, p. 31-56.

Research output: Contribution to journalArticle

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