Turán's theorem inverted


In this note we complete an investigation started by Erdo{double acute}s in 1963 that aims to find the strongest possible conclusion from the hypothesis of Turán's theorem in extremal graph theory. Let Kr+ (s1, ..., sr) be the complete r-partite graph with parts of sizes s1 ≥ 2, s2, ..., sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove that:. For all r ≥ 2 and all sufficiently small c > 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a Kr+ (⌊ c ln n ⌋, ..., ⌊ c ln n ⌋, ⌈ n1 - sqrt(c) ⌉). We also give a corresponding stability theorem and two supporting results of wider scope. © 2009 Elsevier B.V. All rights reserved.

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Discrete Mathematics