Eckhoff’s Problem on Convex Sets in the Plane - Jobson, Adam S.; Kézdy, André E.; Lehel, Jenő (The Electronic Journal of Combinatorics, 2021)
Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner (p,q)-problems as follows. Let F be a finite family of convex sets in the plane and let m1 be an integer. If among every m+22 members of 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 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.