Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices

Abstract

A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl-Ray-Chaudhuri-Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most $\binom{n}{s}$ sets if it is uniform and at most $\sum_{i=0}^s \binom{n}{i}$ sets if it is nonuniform. They also prove modular versions of these results. We consider the following extension of these problems. Call the set systems $\mathcal{A}_1,\ldots,\mathcal{A}_k$ {\em L-cross-intersecting} if for every pair of distinct sets A,B with $A \in \mathcal{A}_i$ and $B \in \mathcal{A}_j$ for some $i \neq j$ the intersection size $|A \cap B|$ lies in $L$. For any k and for n > n 0 (s) we give tight bounds on the maximum of $\sum_{i=1}^k |\mathcal{A}_i|$. It is at most $\max\, \{k\binom{n}{s}, \binom{n}{\lfloor n/2 \rfloor}\}$ if the systems are uniform and at most $ \max\, \{k \sum_{i=0}^s \binom{n}{i} , (k-1) \sum_{i=0}^{s-1} \binom{n}{i} + 2^n\}$ if they are nonuniform. We also obtain modular versions of these results. Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system $\mathcal{A}$ on [n] is a matrix M with rows indexed by $\mathcal{A}$ and columns by the subsets of [n] of size at most s, where if $A \in \mathcal{A}$ and $B \subset [n]$ with $|B| \leq s$, we define M AB to be 1 if $B \subset A$ and 0 otherwise. Our bound generalizes the well-known result that if $|\mathcal{A}| < 2^{s+1}$, then M has full rank $|\mathcal{A}|$. In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed-Muller codes.