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**Zentralblatt MATH**

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

**Zbl.No: ** 565.05042

**Autor: ** Erdös, Paul; Simonovits, M.

**Title: ** Cube-supersaturated graphs and related problems. (In English)

**Source: ** Progress in graph theory, Proc. Conf. Combinatorics, Waterloo/Ont. 1982, 203-218 (1984).

**Review: ** [For the entire collection see Zbl 546.00007.]

For a graph H and an integer n \geq 1, let ex(n,H) denote the maximum number of edges of a graph G on n vertices that contains no copy of H. This paper considers the following conjecture: for every graph H with v vertices and e edges and for every c > 0, there is a constant d > 0 such that every graph G on n vertices with E \geq (1+c)ex(n,H) edges contains at least d· E^{e}/n^{2e-v} copies of H. This conjecture holds for every nonbipartite H by the results of the authors [Combinatorica 3, 181-192 (1983; Zbl 529.05027)]. (See also [*P. Frankl* and *V. Rödl*, Hypergraphs do not jump, ibid. 4, 149-159 (1984)].) If true, the conjecture is best possible. This interesting paper proves the conjecture and some related results for various special cases.

**Reviewer: ** N.Alon

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

00A07 Problem books

**Keywords: ** supersaturated graphs; Turán-type problems

**Citations: ** Zbl 546.00007; Zbl 529.05027

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