Forbidding just one intersection for short integer sequences

TL;DR

This paper investigates the maximum size of subfamilies of sequences over an alphabet that avoid pairs differing in exactly one coordinate, extending recent results using advanced combinatorial techniques.

Contribution

It provides a new bound for the Erdős–Sós forbidden intersection problem for sequences, applicable when alphabet size and sequence length are polynomial in the parameter t.

Findings

Established bounds for large alphabet and sequence length cases

Extended previous results in forbidden intersection problems

Applied novel combinatorial methods to solve the problem

Abstract

In this paper, we study the famous Erd\H{o}s--S\'os forbidden intersection problem for words over an alphabet of size $m$: what is the maximal size of a subfamily $\mathcal{F}$ of $[m]^n$ that does not contain two vectors $x, y$ coinciding on exactly $t - 1$ coordinates? We answer this question provided $m \ge \operatorname{poly}(t)$ and $n \ge \operatorname{poly}(t)$ for some polynomial function $\operatorname{poly}(\cdot)$ of $t$, greatly extending the recent result of Keevash, Lifshitz, Long and Minzer. Our proof combines some of the recently developed methods in extremal combinatorics, including the spread approximation technique of Kupavskii and Zakharov and the hypercontractivity approach developed in a series of works by Keevash, Keller, Lifshitz, Long, Marcus and Minzer.