Fractional discrete Helly for pairs in a family of boxes

2025/03/15 by Eom, Taehyun, Kim, Minki, Lee, Eon · 1 citation
#52A35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.11997

Abstract

Given a point set S in 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 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.

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