Given $n$ sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\cdots\times X_n$ 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.
@article{10_37236_10291,
author = {Peter Frankl and J\'anos Pach},
title = {On well-connected sets of strings},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/10291},
zbl = {1486.05330},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10291/}
}
TY - JOUR
AU - Peter Frankl
AU - János Pach
TI - On well-connected sets of strings
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10291/
DO - 10.37236/10291
ID - 10_37236_10291
ER -
%0 Journal Article
%A Peter Frankl
%A János Pach
%T On well-connected sets of strings
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10291/
%R 10.37236/10291
%F 10_37236_10291
Peter Frankl; János Pach. On well-connected sets of strings. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10291
