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

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

Forbidden sparse intersections

Let $n$ be a positive integer, let $0<p\leqslant p'\leqslant \frac{1}{2}$, and let $\ell \leqslant pn$ be a nonnegative integer. We prove that if $\mathcal{F},\mathcal{G}\subseteq \{0,1\}^n$ are two families whose cross intersections forbid $\ell$ -- that is, they satisfy $|A\cap B|\neq \ell$ for every $A\in\mathcal{F}$ and every $B\in\mathcal{G}$ -- then, setting $t:=\min\{\ell,pn-\ell\}$, we have the subgaussian bound $$\mu_p(\mathcal{F})\, \mu_{p'}(\mathcal{G})\leqslant 2\exp\Big( - \frac{t^2}{58^2\,pn}\Big),$$ where $\mu_p$ and $\mu_{p'}$ denote the $p$-biased and $p'$-biased measures on $\{0,1\}^n$ respectively

Forbidden vector-valued intersections

We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk's problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-R\"odl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai's conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures.Comment: 40 page