Intersection problems and a correlation inequality for integer sequences

TL;DR

This paper investigates intersection problems for codeword collections over finite alphabets, providing near-complete solutions for cases with three or more symbols using a correlation inequality as a key tool.

Contribution

It offers a nearly complete characterization of maximum code sizes under intersection constraints for alphabets of size three or more, advancing understanding of these combinatorial problems.

Findings

Derived a correlation inequality crucial for the analysis.

Provided near-complete solutions for s ≥ 3.

Addressed longstanding open problem for s=2.

Abstract

Let us consider a collection $\mathcal G$ of codewords of length $n$ over an alphabet of size $s$. Let $t_1,\ldots, t_s$ be nonnegative integers. What is the maximum of $|\mathcal G|$ subject to the condition that any two codewords should have at least $t_i$ positions where both have letter $i$ ($1\le i\le s$). In the case $s=2$ it is a longstanding open question. Quite surprisingly, we obtain an almost complete answer for $s\ge 3$. The main tool is a correlation inequality.