We describe several variants of the norm-graphs introduced by Kolla r, Ro nyai, and Szabo and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 1 2 n 5Â3 edges, containing no copy of K 3, 3 , thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K 3, 3 is (1+o(1)) k 3 . This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose so-called dual shatter function is O(m t ) and whose discrepancy is 0(n 1Â2&1Â2t -log n). This settles a problem of Matous ek.

For every t > 1 and positive n we construct explicit examples of graphs G with IV(G)I =n, 2 ] IE(G)I > c t .n -• which do not contain a complete bipartite graph Kt,t!+l. This establishes the exact order of magnitude of the qSar~n numbers ex(n, Kt,s) for any fixed t and all s > t! + 1, improving over the previous probabilistic lower bounds for such pairs (t,s). The construction relies on elementary facts from commutative algebra.

ABSTRACT:We study biased Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that Maker, occupying one edge in each of his turns, can build a spanning tree, even if Breaker occupies bedges in each turn. This improves a result of Beck, and is asymptotically best possible as witnessed by the Breaker-strategy of Chvátal and Erdős. We also give a strategy for Maker to occupy a graph with minimum degree c (where c = c(n) is a slowly growing function of n) while playing against a Breaker who takes b ≤ (1 − o(1)) · n ln n edges in each turn. This result improves earlier bounds by Krivelevich and Szabó. Both of our results support the surprising random graph intuition: the threshold bias is asymptotically the same for the game played by two "clever" players and the game played by two "random" players.

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