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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 526.05031

**Autor: ** Erdös, Paul; Hajnal, András; Sos, Vera T.; Szemeredi, E.

**Title: ** More results on Ramsey-Turán type problems. (In English)

**Source: ** Combinatorica 3, 69-81 (1983).

**Review: ** In [Combinat. Struct. Appl., Proc. Calgary Internat. Conf. Calgary 1969, 407-410 (1970; Zbl 253.05145)] *V.T.Sós* raised a general scheme of new problems that can be considered as common generalizations of the problems treated in the classical results of Ramsey and Turán. This paper is a continuation of a sequence of papers on this subject.

One of the main results is the following: Given k \geq 2 and \epsilon > 0, let G_{n} be a sequence of graphs of order n size at least (½)**(**\frac{3k-5}{3k-2}+\epsilon**)**n^{2} edges such that the cardinality of the largest independent set in G_{n} is o(n). Let H be any graph of arboricity at most k. Then there exists an n_{0} such that all G_{n} with n > n_{0} contain a copy of H. This result is best possible in the case H = K_{2k}.

**Reviewer: ** L.Lesniak

**Classif.: ** * 05C35 Extremal problems (graph theory)

05C55 Generalized Ramsey theory

05C05 Trees

**Keywords: ** arboricity; sequence of graphs; largest independent set

**Citations: ** Zbl.253.05145

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