Type: Article

About an Erdos-Grünbaum conjecture concerning piercing of non-bounded convex sets

Montejano, Amanda; Montejano, Luis; Roldán-Pensado, Edgardo; Soberón, Pablo;

Abstract:

A family F with at least p closed convex sets in Rd satisfies the (p,q)-property if among any p sets in the family there are q of them with a point in common. The piercing number, ?(F), is the minimum cardinality of a set S?Rd such that every set in the family contains at least one point from S. If there is no finite set intersecting the whole family, put ?(F)=?. A well-known result of N. Alon and D. J. Kleitman [Adv. Math. 96 (1992), no. 1, 103–112; MR1185788] states that for any two positive integers p,q, with p?q?d+1, there is a constant c=c(p,q,d) such that every finite family F of closed convex sets in Rd with the (p,q)-property satisfies ?(F)?c. Denote by ?(p,q,d) the smallest constant c with this property. The main result of the paper is the following: Theorem. Let p?q?d+1 be positive integers. (i) If q?p?q+(d+1), and F is a family of closed convex sets in Rd containing at least p?q+1 bounded members and satisfying the (p,q)-property, then ?(F)??(q?1,d,d?1)?(p,q,d)+p?q+1. (ii) If q<p?q+(d+1), then there is a family F of closed convex sets in Rd, containing infinitely many bounded members, satisfying the (p,q)-property and such that ?(F)=?.