Extremal set systems with restricted k-wise intersections

A large variety of problems and results in Extremal Set Theory deal with estimates on the size of a family of sets with some restrictions on the intersections of its members. Notable examples of such results, among others, are the celebrated theorems of Fischer, RayChaudhuri–Wilson and Frankl–Wilson on set systems with restricted pairwise intersections. These also can be considered as estimates on binary codes with given distances. In this paper we obtain the following extension of some of these results when the restrictions apply to k-wise intersections, for k42: Let L be a subset of non-negative integers of size s and let k42: A family F of subsets of an n-element set is called k-wise L-intersecting if the cardinality of the intersection of any k distinct members in F belongs to L:We prove that, for any fixed k and s and sufficiently large n; the size of every k-wise L-intersecting family is bounded by jFjp þ s 1 s þ 1 n s þ X ips 1 n i : ARTICLE IN PRESS Corresponding author. E-mail addresses: [email protected] [email protected] (Z. F . uredi), [email protected] (B. Sudakov). Research supported in part by NSF grant DMS 01-40692 and OTKA Grant T 032452. Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. 0097-3165/$ see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2003.10.008 This result is asymptotically best possible. In addition, we show that for an extremal k-wise L-intersecting family, L consists of s consecutive integers. Our proof combines tools from linear algebra with some combinatorial arguments. r 2003 Elsevier Inc. All rights reserved.