More Sets, Graphs and Numbers: A Salute to Vera Sòs and by Ervin Gyori, Gyula O.H. Katona, László Lovász

By Ervin Gyori, Gyula O.H. Katona, László Lovász

This quantity honours the eminent mathematicians Vera Sos and Andras Hajnal. The e-book comprises survey articles reviewing classical theorems, in addition to new, cutting-edge effects. additionally awarded are leading edge expository learn papers with new theorems and proofs within the region of the classical Hungarian topics, like extremal combinatorics, colours, combinatorial quantity thought, and so on. The open difficulties and the newest leads to the papers are guaranteed to encourage additional research.

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Krivelevich, On th e edge distribution in t riangle-free graphs, J. Combinatorial Th eory, Ser. B, 63 (1995), 245-260 . [10] J. E. Littlewood, On bound ed bilinear forms in an infinite number of variables, Quart . J. Math . Oxford, 1 (1930), 164-174 . [11] V. T . S6s, Irregularities of partitions: Ramsey theory, uniform distribution , in: Surveys in Combinatorics (Southampton , 1983),201-246, London Math . Soc. Lecture Note Ser. , 82, Cambridge Univ. Pr ess, Cambridge-New York, 1983. [12] S. J .

RESULTS AND PROBLEMS ON COMPLETE BIPARTITE GRAPHS The famous Zarankiewicz 's Crossing Number Conjecture or Turan's Brick Factory Problem is as follows: (12) cr (Kp,q) = l~J lp; J l~J lq; J. 1 1 Kleitman showed that (12) holds for q :::; 6 [91 and also proved that the smallest counterexample to the Zarankiewicz 's conjecture must occur for odd p and q. Woodall used elaborate computer search to show that (12) holds for f{7,7 and f{7,9' Thus, the smallest unsettled instances of Zarankiewicz's conjecture are K 7,1l and Kg,g.

We can now state t he first result of th e section. Theorem 8. Let G be th e complete k-uniform hypergraph of order n with edge-weighting w such th at L w(e) = 0 an d L Iw(e)1 = G). Th en disc" (G) disc- (G) ~ T 2 14k n k+ l . We sha ll need three lemm as. ) Lemma 9. If G is a com plete k -uniform hypergraph with edge-weighting wand disc (G) ::; M th en [ or disj oint subsets X , Y oi V(G) and 0 ::; i ::; k, Proof. Let Z be a random subset of X , where each vertex is chosen independently with probability p.

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