Systems of Overlap Representation for Families of Intervals

Abstract

Two sets overlap if they intersect and neither contains the other. Given a family $\mathcal {F}$ of sets, a system of overlap representation (SOR) for $\mathcal {F}$ assigns to each set $X\in \mathcal {F}$ a subset $S_X$ of X, called its representative set, so that the representative sets chosen for any two overlapping members of $\mathcal {F}$ intersect. Let $\mathcal {F}_n$ be the family of intervals of integers contained in $\{1,\ldots ,n\}$, and let f(n) be the minimum of the maximum size of the sets in an SOR of $\mathcal {F}_n$. We prove

$$(1-o(1))\lg (n-1)<f(n)\le 2\lg (n-1).$$