Eckhoff's problem on convex sets in the plane

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 m >= 1 be an integer. If among every (m+2 2) members of F all but at most m 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 Szemeredi 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.