The $(p,q)$ property in families of $d$-intervals and $d$-trees

TL;DR

This paper proves new bounds on the number of points needed to pierce families of $d$-intervals and subgraphs of bounded tree-width that satisfy the $(p,q)$ property, extending previous results and establishing asymptotic sharpness.

Contribution

It extends existing piercing number bounds to families of $d$-intervals and subgraphs of bounded tree-width for general $p$ and $q$, and introduces sharp bounds for the fractional piercing number.

Findings

Piercing number bounded by $O(d^{q/(q-1)})$ for families satisfying the $(p,q)$ property.

Similar bounds established for subgraphs of trees or graphs with bounded tree-width.

Upper bound of $O(d^{1/(p-1)})$ on fractional piercing number, proven to be asymptotically sharp.

Abstract

Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$ property can be pierced by $O(d^{\frac{q}{q-1}})$ points, with constants depending only on $p$ and $q$. This extends results of Tardos, Kaiser and Alon for the case $q=2$, and of Kaiser and Rabinovich for the case $p=q=\lceil log_2(d+2) \rceil$. We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consisting of at most $d$ connected components, extending results of Alon for the case $q=2$. Finally, we prove an upper bound of $O(d^{\frac{1}{p-1}})$ on the fractional piercing number in families of $d$-intervals satisfying the $(p,p)$ property, and show that this bound is asymptotically sharp.