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

Abstract: A set in $\mathbb{R}<sup>d$ 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 $\mathrm{HD}<em>d<sup>{(s)}(p,q)$ 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 $\mathrm{HD}_d<sup>{(s)}(p,q)$ 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., $\mathrm{HD}_1<sup>{(s)}(p,q)$). In particular, we prove the following: (i) For every $d\ge2$, $s \geq 1$ and $δ&gt;0$, if $p&gt;q$ and $q\ge C_d\log(e sp)$, then $$\mathrm{HD}_d^{(s)}(p,q) \le p-q+1 + O</em>{d,δ}((s \cdot \tfrac{p}{q} \cdot \log \tfrac{esp}{q})<sup>{ρ_d+δ}),$$ where $ρ_d&lt;d$ 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 $$q\&gt;2$$. (iii) For any fixed $s$, there are an integer $κ_s\in{s,\ldots,2s}$ and constants $C_s,p_s&gt;0$ 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 $s \geq 1$, $$\mathrm{HD}_3^{(s)}(p,4) \geq sp<sup>{2-o(1)}$$. Already for families of convex sets, this significantly improves the best known lower bound on $\mathrm{HD}_d<sup>{(1)}(p,d+1)$, for all $d \geq 3$.

• The paper develops lexicographic certificate counting and weak-net methods to improve general-dimensional upper bounds for unions of at most $s$ convex sets, reducing dependence on $s$ compared with component reduction.
• For unions of intervals, it proves near-tight bounds $p-q+s le HD_1^{(s)}(p,q) le p-q+2s+1$ in a broad logarithmic range and establishes two-value concentration for fixed $s$.
• The paper raises convex lower bounds to $p^{2-gamma}$ in dimensions $dge3$, amplifies them to $s$-convex sets, and derives the large-$q$ obstruction $p-q+d(s-1)+1$.

Overview and context

The Hadwiger–Debrunner $(p,q)$ problem asks for the smallest number of points needed to pierce a finite family of sets in $\mathbb{R}^d$ satisfying the $(p,q)$-property: among every $p$ members, some $q$ intersect. Alon and Kleitman's $(p,q)$-theorem resolved the qualitative question for convex sets, but quantitative bounds remain poorly understood. This paper by Keller and Smorodinsky extends the study to $s$-convex sets—unions of at most $s$ convex sets—a class with no finite Helly number (Larman), for which Alon and Kalai established existence of bounded transversals with very loose bounds. Writing $HD_d^{(s)}(p,q)$ for the least piercing number guaranteed over all finite families of $s$-convex sets with the $\mathbb{R}^d$0-property, the authors obtain substantially improved upper bounds in general dimension, near-tight estimates for unions of intervals on the line, and new superlinear lower bounds that apply already to ordinary convex sets.

Upper bounds in general dimension

The central technical device is a lexicographic certificate count. For a family $\mathbb{R}^d$1 of $\mathbb{R}^d$2-convex sets with maximum depth $\mathbb{R}^d$3, the number $\mathbb{R}^d$4 of intersecting $\mathbb{R}^d$5-tuples satisfies $\mathbb{R}^d$6: each intersecting tuple is certified by the lexicographic minimum of its intersection, which by Helly's theorem is determined by at most $\mathbb{R}^d$7 convex components. Compared to the naive reduction to all $\mathbb{R}^d$8 convex components, this saves an essential factor $\mathbb{R}^d$9 in the final bounds.

Combining this count with de Caen's Turán-type theorem yields a depth estimate: any family satisfying the $(p,q)$0-property contains a point lying in at least $(p,q)$1 members. Via the LP-duality lemma and weak $(p,q)$2-nets (using Rubin's improved exponent $(p,q)$3), this gives:

$(p,q)$4

For large $(p,q)$5, the $(p,q)$6-dichotomy bootstrapping of Keller–Smorodinsky–Tardos reduces the problem to scale $(p,q)$7, yielding the stronger estimate

$(p,q)$8

valid whenever $(p,q)$9. The certificate count also directly implies a fractional Helly theorem for $p$0-convex sets: if at least an $p$1 fraction of the $p$2-tuples intersect, some point lies in at least $p$3 members—one power of $p$4 better than component reduction—and a colorful analogue follows from a pigeonhole argument over omitted colors.

The one-dimensional case

On the line, where weak $p$5-nets have optimal size $p$6, the same machinery gives $p$7 for all parameters—recovering Zerbib's exponent of $p$8 with explicit linear dependence on $p$9 (improving her polynomial dependence)—and, when $q$0,

$q$1

The main one-dimensional result combines these estimates with iterated dichotomy bootstrapping and Zerbib's diagonal bound $q$2 as the terminal case. It establishes the first near-tight range for $q$3: whenever $q$4,

$q$5

This is the first broad range with matching linear dependence on $q$6 in both upper and lower bounds for $q$7. The lower bound construction takes complements of tiny open "holes" in $q$8 chosen via a compactness argument so that any $q$9 holes cannot cover the interval while every set of $(p,q)$0 points lies in some hole; disjoint intervals outside $(p,q)$1 supply the additive $(p,q)$2 term.

A striking refinement holds for fixed $(p,q)$3. Since the diagonal numbers $(p,q)$4 form a nonincreasing integer sequence bounded between $(p,q)$5 and $(p,q)$6, they eventually stabilize at some value $(p,q)$7. A cleanup lemma then shows that once a constant-additive bound is available in the logarithmic range, one dichotomy step lowers the additive constant by one until the diagonal obstruction is reached. Consequently there exist constants $(p,q)$8 such that, for all $(p,q)$9 and $s$0,

$s$1

This two-value concentration occurs even though the exact value of $s$2 remains unknown—the argument does not determine whether $s$3, and improving below $s$4 by this method would require a sharper diagonal estimate.

Lower bounds

Convex case. Using a higher-dimensional general-position construction of Suk and Zeng (point sets in $s$5 with no $s$6 points co-hyperplanar whose largest general-position subset has size $s$7), together with point–hyperplane duality, the paper proves that for every fixed $s$8 and $s$9,

$s$0

for arbitrarily large $s$1. This improves the previously best bound $s$2 of Keller–Smorodinsky to an almost quadratic exponent, resolving the possibility that $s$3 might be quasi-linear in $s$4 in dimensions $s$5.

Amplification to unions. An amplification lemma places $s$6 affine copies of an extremal convex family in disjoint balls and forms $s$7-fold unions; a counting argument shows any transversal must fully pierce at least $s$8 copies. This converts any convex lower bound into an $s$9-convex one with linear gain in $HD_d^{(s)}(p,q)$0, giving $HD_d^{(s)}(p,q)$1.

Large $HD_d^{(s)}(p,q)$2. A direct construction generalizes the interval "holes" argument using the Steiner convex partition theorem: inside a simplex, Steiner partitions into $HD_d^{(s)}(p,q)$3 tiles avoiding any prescribed $HD_d^{(s)}(p,q)$4 points yield almost-full $HD_d^{(s)}(p,q)$5-convex sets, and a compactness/finite-subcover argument produces families showing

$HD_d^{(s)}(p,q)$6

for all $HD_d^{(s)}(p,q)$7. In particular, the nearly exact bound $HD_d^{(s)}(p,q)$8 valid for ordinary convex sets does not extend to $HD_d^{(s)}(p,q)$9-convex sets once $s$0 and $s$1; the correct additive order grows linearly with both components and dimension.

Limitations and open questions

Several gaps remain explicit in the paper. The exact value of the stabilization constant $s$2 is undetermined, and it is unknown whether $s$3. The upper bound $s$4 carries an extra additive $s$5 arising from the exceptional subfamily in the diagonal reduction; removing it requires either a diagonal estimate below $s$6 or an argument avoiding the diagonal case entirely. The general-dimension upper bounds depend on Rubin's weak-net exponent $s$7 and lose a factor $s$8, so they are not tight against the lower bound $s$9. Finally, the lower bound $\mathbb{R}^d$00 in dimension one matches the polylogarithmic-range theorem only up to an additive factor of about $\mathbb{R}^d$01, leaving the precise constant in that regime unresolved.

Conclusion

The paper delivers a coordinated set of quantitative advances for the $\mathbb{R}^d$02-problem on unions of convex sets: lexicographic certificate counting improves general-dimension upper bounds by essentially $\mathbb{R}^d$03 over component reduction; bootstrapping plus Zerbib's diagonal estimate yields matching-order bounds for $\mathbb{R}^d$04-intervals and an unexpected two-value concentration phenomenon for fixed $\mathbb{R}^d$05; and constructions based on general-position point sets and Steiner partitions raise the known lower bounds to $\mathbb{R}^d$06 for $\mathbb{R}^d$07 and establish the dimension-dependent obstruction $\mathbb{R}^d$08 in the large-$\mathbb{R}^d$09 regime.