## 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· Ee/n2e-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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