By using the Szemeredi Regularity Lemma, Alon and Sudakov recently\ud extended the classical Andrasfai-Erdos-Sos theorem to cover general graphs. We\ud prove, without using the Regularity Lemma, that the following stronger statement\ud is true.\ud Given any (r+1)-partite graph H whose smallest part has t vertices, there exists\ud a constant C such that for any given ε>0 and sufficiently large n the following is\ud true. Whenever G is an n-vertex graph with minimum degree\ud δ(G)≥(1 −\ud 3/3r−1 + ε)n,\ud either G contains H, or we can delete f(n,H)≤Cn2−1/t edges from G to obtain an\ud r-partite graph. Further, we are able to determine the correct order of magnitude\ud of f(n,H) in terms of the Zarankiewicz extremal function
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