Induced Complete h-partite Graphs in Dense Clique-less Graphs

It is proven that for every fixed h, a and b, a graph with n vertices and minimum degree at least h−1 h n, which contains no copy of Kb (the complete graph with b vertices), contains at least (1 − o(1)) n ha vertex disjoint induced copies of the complete h-partite graph with a vertices in each color class. Mathematics subject classification numbers: 05C70, 05C55. All graphs considered here are finite, undirected, and have neither loops nor parallel edges. The notation here follows the convention of [4] except where stated otherwise. Many asymptotic embedding results have been proven by the Regularity Lemma of Szemerédi [6] over the years. See [5] for a survey. One of the results in this area, that of Alon and Yuster [2], has particular relevance to the following. Theorem 1 ([2]) For every natural a, h and every > 0 there exists N = N(h, a, ) such that if G is a graph with n > N vertices and minimum degree at least h−1 h n, then G contains at least (1− ) n ha vertex disjoint copies of K(h, a), the complete h-partite graph with a vertices in each color class. ∗Research supported by the Fritz Brann Doctoral Fellowship in Engineering and Exact Sciences.