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

Abstract

The Hadwiger-Debrunner number HD d (p, q) is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in R d that satisfies the (p, q) property. 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 particular, we show that for p, q sufficiently large, HD d (p, q) r ≤ p − q + 1 holds with r = 1 p q 2d p q . Our bound misses the known lower bound for the same piercing number by a factor of less than pq d .Our results use Kalai's Upper Bound Theorem for convex sets, along with the Hadwiger-Debrunner theorem and the recent improved upper bound on HD d (p, q) mentioned above.

Untitled

“…A family of sets has the (p, q) r -property if in its nerve every p vertices span at least r faces with q vertices. This was introduced by Montejano and Soberón [MS11] and further studied by Keller and Smorodinsky [KS18]. Montejano and Soberón proved (among other results)…”

The Haunted Landscape of the Uncanny North: Scott Graham’s Shell (2012) and Gordon Napier’s 1745 (2017)

“…7 The coexistence of multiple narratives inscribed in landscape across time further complicates the 3 triangulation of landscape, identity and national history: '[l]andscape is a record of time', Patricia M. Keller explains, 'it holds history and "remembers" it materially'. 8 To advance Angela Wright's claim that '[g]raves, castles, manuscripts and inscriptions are all warmly contested sites of authenticity and authority' in Scottish Gothic, 9 therefore, I would argue that the north is, aesthetically as well as politically, the disputed territory of Scotland's identity narratives. Beyond the mourning for a nation's territory by its (dispossessed) people, Scotland's northern regions are host to Other invisible histories, which, rather than being 'under the threat of erasure', actively threaten to possess, or even repossess, the landscape that has concealed them.…”