On the Zarankiewicz problem for intersection hypergraphs
Let d and t be fixed positive integers, and let denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erd6s , the number of hyperedges of a d-uniform hypergraph on n vertices that does not contain as a subhypergraph, is nd-1/t(d-1). This bound is not far from being optimal. We address the same problem restricted to intersection hypergraphs of (d 1)-dimensional simplices in Rd. Given an n-element set S of such simplices, let H-d(S) denote the d-uniform hypergraph whose vertices are the elements of S, and a d-tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if H-d(S) does not contain K-t(d) . . t , as a subhypergraph, then its number of edges is O(n) if d = 2, and O(n(d-1)++epsilon) for any e > 0 if d >= 3. This is almost a factor of n better than Erdos's above bound. Our result is tight, apart from the error term E in the exponent. In particular, for d = 2, we obtain a theorem of Fox and Pach , which states that every Kt,t-free intersection graph of n segments in the plane has 0(n) edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses size-sensitive cuttings, a variant of random sampling. (C) 2016 Elsevier Inc. All rights reserved.