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Fractional discrete Helly for pairs in a family of boxes

Taehyun Eom, Minki Kim, Eon Lee

TL;DR

This work addresses fractional S-Helly properties for axis-parallel boxes in $R^d$ by linking densities of pairwise ($2$-tuples) and higher-order intersections. It develops a permutation-based, projection-driven combinatorial framework and employs Ramsey theory and supersaturation to prove that high density of $S$-intersecting pairs forces a large $S$-intersecting subfamily, with a finite-threshold reduction extending to all sizes. Key contributions include a qualitative fractional S-Helly result for pairs, the exact value $N(2)=5$, and a linear bound for the $d$-tuple analogue, thereby connecting Helly-type thresholds with fractional intersection properties in discrete geometric set systems. These results advance understanding of how local intersection patterns control global structure in families of axis-parallel boxes, with potential algorithmic and combinatorial implications.

Abstract

Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Soberón proved a fractional version of Halman's theorem for axis-parallel boxes, showing that every finite family $F$ of axis-parallel boxes in $\mathbb{R}^d$ with positive density of $S$-intersecting $(d+1)$-tuples contains an $S$-intersecting subfamily of size linear in $|F|$. We prove that qualitatively the same conclusion can be achieved if the density of $S$-intersecting pairs is sufficiently large.

Fractional discrete Helly for pairs in a family of boxes

TL;DR

This work addresses fractional S-Helly properties for axis-parallel boxes in by linking densities of pairwise (-tuples) and higher-order intersections. It develops a permutation-based, projection-driven combinatorial framework and employs Ramsey theory and supersaturation to prove that high density of -intersecting pairs forces a large -intersecting subfamily, with a finite-threshold reduction extending to all sizes. Key contributions include a qualitative fractional S-Helly result for pairs, the exact value , and a linear bound for the -tuple analogue, thereby connecting Helly-type thresholds with fractional intersection properties in discrete geometric set systems. These results advance understanding of how local intersection patterns control global structure in families of axis-parallel boxes, with potential algorithmic and combinatorial implications.

Abstract

Given a point set in , a family of sets is -intersecting if its members have a point in common in . Recently, Edwards and Soberón proved a fractional version of Halman's theorem for axis-parallel boxes, showing that every finite family of axis-parallel boxes in with positive density of -intersecting -tuples contains an -intersecting subfamily of size linear in . We prove that qualitatively the same conclusion can be achieved if the density of -intersecting pairs is sufficiently large.

Paper Structure

This paper contains 5 sections, 13 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Intersection patterns of pairwise $S$-intersecting axis-parallel boxes
  3. Proof of Theorem \ref{['thm:main2']}
  4. Finding the optimal value of $N(2)$
  5. Intersection patterns of $d$-tuplewise $S$-intersecting axis-parallel boxes

Key Result

Theorem 1.1

For every positive integer $d$, there is a number $c_d\in(0,1)$ and a function $\beta_d:(c_d,1]\to(0,1]$ such that the following holds: for every $S \subset\mathbb{R}^d$, $\alpha\in(c_d,1]$, and finite family $F$ of axis-parallel boxes in $\mathbb{R}^d$, if there are at least $\alpha\binom{|F|}{2}$

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2: ES24+
  • Theorem 1.3
  • Theorem 1.4: supersaturation
  • proof : Proof of Theorem \ref{['thm:main']} from Theorem \ref{['thm:main2']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 14 more