# AN EXTREMAL PROBLEM IN GRAPH THEORY

@inproceedings{Preston2001ANEP, title={AN EXTREMAL PROBLEM IN GRAPH THEORY}, author={G. B. Preston}, year={2001} }

G(?z; I) will denote a graph of n vertices and 1 edges. Let fO(lz, K) be the smallest integer such that there is a G (n; f,, (n, k)) in which for every set of K vertices there is a vertex joined to each of these. Thus for example fO(3, 2) = 3 since in a triangle each pair of vertices is joined to a third. It can readily be checked that f,(4, 2) = 5 (the extremal graph consists of a complete 4-gon with one edge removed). In general we will prove: Let n > k, and

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