Disjoint pairs in set systems and combinatorics of low rank matrices

TL;DR

This paper resolves longstanding problems in combinatorics and matrix theory, establishing optimal bounds on disjoint pairs in set families and low-rank matrices with many zero entries, with implications for the log-rank conjecture.

Contribution

It provides complete solutions to the maximum disjoint pairs problem in set families and proves optimal bounds for low-rank matrices with many zeros, advancing understanding in combinatorics and matrix theory.

Findings

Resolved the maximum disjoint pairs problem in set families.

Proved a strong quantitative version of a Lovett-related conjecture.

Established optimal bounds for zero submatrices in low-rank matrices.

Abstract

We study and solve several problems in two closely related settings: set families in $2^{[n]}$ with many disjoint pairs of sets and low rank matrices with many zero entries. - More than 40 years ago, Daykin and Erd\H{o}s asked for the maximum number of disjoint pairs of sets in a family $F\subseteq 2^{[n]}$ of size $2^{(1/2+\delta)n}$ and conjectured it contains at most $o(|F|^2)$ such pairs. This was proven by Alon and Frankl in 1985. In this paper we completely resolve this problem, proving an optimal dependence of the number of disjoint pairs on the size of family $F$. We also prove the natural variant of the Daykin-Erd\H{o}s conjecture in which disjoint pairs are replaced by pairs with intersection $\lambda\neq 0$. - Motivated by a conjecture of Lovett related to the famous log-rank conjecture, Singer and Sudan asked to show that for two families $A, B \subseteq 2^{[n]}$ with a positive constant fraction of set pairs $(a,b)\in A\times B$ being disjoint, there are $R\subset A$ and $S\subset B$ such that all set pairs $(r, s)\in R\times S$ are disjoint, and $|R|\geq 2^{-O(\sqrt{n})}|A|$ and $|S|\geq 2^{-O(\sqrt{n})}|B|$. We prove this conjecture in a strong quantitative form. - We prove the following generalizations of the best known bounds for the log-rank conjecture. If $M$ is an $n\times n$ non-negative integer matrix of rank $r$ in which the average of the entries is $\varepsilon\leq 1/2$, then $M$ contains an all-zero submatrix of size at least $2^{-O(\sqrt{\varepsilon r})}n$. Unlike the known bounds for the log-rank conjecture, this result is optimal. Moreover, using similar methods, we also prove that any $n\times n$ matrix of rank $r$ with entries from $\{0,\dots,t\}$ contains a constant submatrix of size at least $2^{-O(t\sqrt{r})}n$. Our proofs use probabilistic, entropy and discrepancy methods and explore connections to additive combinatorics and coding theory.