Piercing Numbers for Balanced and Unbalanced Families

Abstract

Given a finite family F of convex sets in R d , we say that F has the (p, q) r property if for any p convex sets in F there are at least r q-tuples that have nonempty intersection. The piercing number of F is the minimum number of points we need to intersect all the sets in F . In this paper we will find some bounds for the piercing number of families of convex sets with (p, q) r properties.

The geometry and combinatorics of discrete line segment hypergraphs

“…Like many other hypergraph families in the literature [6,4,37], r-uniform hypergraphs are linear, meaning any two edges intersect in at most one vertex. It is worth noting that in general, the problem of deciding whether an r-uniform hypergraph H is k-colorable is NP-hard, even in the case where H is linear [9,34,39].…”

On transversal and 2-packing numbers in uniform linear systems

“…A subset T ⊆ P is a transversal (also called vertex cover or hitting set in many papers, as example [7,9,11,12,14,[16][17][18][19][20][21]) of (P, L) if for any line l ∈ L satisfies T ∩ l = ∅. The transversal number of (P, L), denoted by τ (P, L), is the smallest possible cardinality of a transversal of (P, L).…”

On piercing numbers of families satisfying the (p,q) property

“…HD d (p, q) r is defined as the maximal piercing number taken over all families that satisfy the (p, q) r property. The main result of [9] is:…”

“…Hadwiger and Debrunner showed that HD d (p, q) ≥ p − q + 1 for all q, and equality is attained for q > d−1 d p+1. Almost tight upper bounds for HD d (p, q) for a 'sufficiently large' q were obtained recently using an enhancement of the celebrated Alon-Kleitman theorem, but no sharp upper bounds for a general q are known.In [9], Montejano and Soberón defined a refinement of the (p, q) property: F satisfies the (p, q) r property if among any p elements of F , at least r of the q-tuples intersect. They showed that HD d (p, q) r ≤ p − q + 1 holds for all r > p q − p+1−d q+1−d ; however, this is far from being tight.In this paper we present improved asymptotic upper bounds on HD d (p, q) r which hold when only a tiny portion of the q-tuples intersect.…”

“…In [9], Montejano and Soberón defined a refinement of the (p, q) property: F satisfies the (p, q) r property if among any p elements of F , at least r of the q-tuples intersect. They showed that HD d (p, q) r ≤ p − q + 1 holds for all r > p q − p+1−d q+1−d ; however, this is far from being tight.…”