On well-connected sets of strings

Abstract

Given $n$ pairwise disjoint sets $X_1,\dots ,X_n$, we call the elements of $S=X_1\times \dots \times X_n$ 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 $v^{\prime }\in W$ 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.

arXiv comment6 pages