# On Well-Connected Sets of Strings

### Abstract

Given $n$ sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\cdots\times X_n$ s*trings*. 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.