On well-connected sets of strings

TL;DR

The paper proves a conjecture that large enough sets of strings formed from disjoint sets always contain a well-connected subset, with a tight bound on the size needed.

Contribution

It establishes the exact size threshold for sets of strings to guarantee the existence of a well-connected subset, confirming a conjecture by Wu and Xiong.

Findings

Proved the conjecture on well-connected subsets.

Established the tight bound for the size of such sets.

Demonstrated the existence of well-connected subsets in large string sets.

Abstract

Given $n$ pairwise disjoint sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\ldots\times X_n$ strings. A nonempty set of strings $W\subseteq S$ is said to be well-connected if for every $v\in W$ and for every $i\, (1\le i\le n)$, there is another element $v'\in W$ which differs from $v$ only in its $i$th coordinate. We prove a conjecture of Yaokun Wu and Yanzhen Xiong by showing that every set of more than $\prod_{i=1}^n|X_i|-\prod_{i=1}^n(|X_i|-1)$ strings has a well-connected subset. This bound is tight.