New Quantitative Bounds for the $(p,q)$-Theorem for Unions of Convex Sets

Abstract

A set in $\mathbb{R}^d$ is $s$-convex if it is the union of at most $s$ convex sets. A family $F$ satisfies the $(p,q)$ property if among any $p$ sets in $F$, some $q$ intersect. Let $\mathrm{HD}_d^{(s)}(p,q)$ be the minimum number of points needed to pierce a finite family of $s$-convex sets that satisfies the $(p,q)$-property. Alon and Kalai (1995) proved that $\mathrm{HD}_d^{(s)}(p,q)$ exists for any $p \geq q \geq d+1$ and any $s \geq 1$, but the quantitative bounds they obtained are very loose. We present several improved upper and lower bounds, for a general $d$ and for $s$-intervals of the line (i.e., $\mathrm{HD}_1^{(s)}(p,q)$). In particular, we prove the following: (i) For every $d\ge2$, $s \geq 1$ and $δ>0$, if $p>q$ and $q\ge C_d\log(e sp)$, then $\mathrm{HD}_d^{(s)}(p,q) \le p-q+1 + O_{d,δ}((s \cdot \tfrac{p}{q} \cdot \log \tfrac{esp}{q})^{ρ_d+δ}),$ where $ρ_d<d$ is the exponent in the weak epsilon-net theorem of Rubin (2022). (ii) For $s \geq 1$, $p \geq q \geq 2$ and $q\ge C_0s\log(2s)\log(ep)$, $p-q+s \leq \mathrm{HD}_1^{(s)}(p,q) \leq p-q+2s+1$. This result provides the first near-tight estimate for $\mathrm{HD}_1^{(s)}(p,q)$ for $q>2$. (iii) For any fixed $s$, there are an integer $κ_s\in\{s,\ldots,2s\}$ and constants $C_s,p_s>0$ such that, whenever $p\ge p_s$ and $q\ge C_s\log(ep)$, $ \mathrm{HD}_1^{(s)}(p,q)\in\{p-q+κ_s,\;p-q+κ_s+1\}. $ Interestingly, this two-value concentration result holds, although the exact value of the threshold remains unknown. (iv) For any $s \geq 1$, $\mathrm{HD}_3^{(s)}(p,4) \geq sp^{2-o(1)}$. Already for families of convex sets, this significantly improves the best known lower bound on $\mathrm{HD}_d^{(1)}(p,d+1)$, for all $d \geq 3$.