An Erdős-Ko-Rado Theorem for Tilings

Abstract

We prove an Erdős-Ko-Rado type extremal result for tilings of a $1\times n$ chessboard by tiles whose lengths belong to a set $\hat{I}›$. Two tilings are said to intersect if they contain a tile spanning the same set of squares. We prove that if $1\in \hat{I}›$, then the maximum size of an intersecting family of tilings is attained by the set of all tilings containing a fixed singleton tile at one of its ends. This result generalizes a theorem of Butler, Horn and Tressler, which is equivalent to the case $\hat{I}›=\{1,2\}$.