A purely combinatorial proof of the Hadwiger Debrunner \((p,q)\) conjecture (Q1378536)

Summary: A family of sets has the \((p,q)\) property if among any \(p\) members of the family some \(q\) have a nonempty intersection. The authors have proved that for every \(p \geq q \geq d+1\) there is a \(c=c(p,q,d) < \infty\) such that for every family \({\mathcal F}\) of compact, convex sets in \(R^d\) which has the \((p,q)\) property there is a set of at most \(c\) points in \(R^d\) that intersects each member of \({\mathcal F}\), thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result.