Eckhoff's problem on convex sets in the plane (Q2049618)

Summary: Eckhoff proposed a combinatorial version of the classical Hadwiger-Debrunner \((p,q)\)-problems as follows. Let \(\mathcal{F}\) be a finite family of convex sets in the plane and let \(m\geqslant 1\) be an integer. If among every \(\binom{m+2}{2}\) members of \(\mathcal{F}\) 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 \(\mathcal{F}\). 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.