N ov 2 00 7 Turán ’ s theorem inverted

Let K+ r (s1, . . . , sr) be the complete r-partite graph with parts of size 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: (A) For all r ≥ 2 and all sufficiently small ε > 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ( ⌊c lnn⌋ , . . . , ⌊c lnn⌋ , ⌈ n1− √ c ⌉) . (B) For all r ≥ 2, there exists c > 0 such that every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r (⌊c lnn⌋ , . . . , ⌊c lnn⌋) . These assertions extend results of Erdős from 1963. We also give corresponding stability results