On the Zarankiewicz Problem for Intersection Hypergraphs

摘要

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 Erdoes, 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 - l)-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+∈) for any ∈ ＞ 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 [8], which states that every K_(t, 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.