On the Zarankiewicz Problem for Intersection Hypergraphs

Let d and t be fixed positive integers, and let K-t(d),..., t 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 Erd. os [7], the number of hyperedges of a d-uniform hypergraph on n vertices that does not contain K-t(d),..., t as a subhypergraph, is n(d-) 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 R-d. 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 epsilon > 0 if d >= 3. This is almost a factor of n better than Erd. os's above bound. Our result is tight, apart from the error term epsilon in the exponent. In particular, for d = 2, we obtain a theorem of Fox and Pach [8], which states that every Kt, t-free intersection graph of n segments in the plane has O(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. We demonstrate the flexibility of this technique by extending the proof of the planar version of the theorem to intersection graphs of x-monotone curves.