Norm-Graphs: Variations and Applications

Abstract

We describe several variants of the norm-graphs introduced by Kollár, Rónyai, and Szabó 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 $\text{1}\text{2}$n^5/3 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)) k³. 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 Ω(n^1/2−1/2t $\text{log}\text{n}$). This settles a problem of Matoušek.