Eckhoff's Problem on Convex Sets in the Plane

Adam S Jobson,André E Kézdy,Jenő Lehel

doi:10.37236/9978

Adam S Jobson, André E Kézdy + Show 1 more

Open Access

https://doi.org/10.37236/9978

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Abstract

Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner $(p,q)$-problems as follows. Let ${\cal F}$ be a finite family of convex sets in the plane and let $m\geqslant 1$ be an integer. If among every ${m+2\choose 2}$ members of ${\cal F}$ all but at most $m-1$ members have a common point, then there is a common point for all but at most $m-1$ members of ${\cal F}$. The claim is an extension of Helly's theorem ($m=1$). The case $m=2$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemerédi and Petruska concerning $3$-uniform hypergraphs. This conjecture is still open in general; its solution for a few special cases answers Eckhoff's problem for $m=3,4$. A new proof for the case $m=2$ is also presented.

Highlights

The subject of this note is a combinatorial version of the classical Hadwiger–Debrunner (p, q)-problems proposed by Eckhoff [2]
M+2 2 be integers, and let F be a family of at least k convex sets in R2
Observe that by Helly’s theorem, a family F of k convex sets in R2 has the ∆(m)property if and only if the 3-uniform intersection hypergraph H defined by F has clique number ω(H) k − m + 1

Summary

Introduction

The subject of this note is a combinatorial version of the classical Hadwiger–Debrunner (p, q)-problems proposed by Eckhoff [2] (see [1]). We restate Eckhoff’s conjecture using this notation. Eckhoff’s conjecture follows from an old conjecture due to Szemeredi and Petruska [10]. The Szemeredi-Petruska conjecture, as reformulated by Lehel and. Tuza [11, Problem 18.(a)] states that m+2 2 is the maximum order of a 3-uniform τ -critical hypergraph with transversal number m. Eckhoff’s conjecture becomes equivalent to a particular instance of a general extremal hypergraph problem (Theorem 6). The. Szemeredi-Petruska conjecture is verified for m = 2, 3, 4 (see [7]) using the concept of 3-. Uniform τ –critical hypergraphs, cross-intersecting set-pair systems, and τ -critical graphs; this solves Eckhoff’s problem for m = 3, 4, with a new proof for m = 2 (Corollary 7).

Convex hypergraphs

Eckhoff ’s problem and τ -critical hypergraphs

Concluding remarks