Transversals and Colorings of Simplicial Spheres

Motivated from the surrounding property of a point set in ℝ^d introduced by Holmsen, Pach, and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial d -spheres, we provide two infinite constructions. The first construction gives infinitely many (d+1) -dimensional simplicial polytopes with the transversal ratio exactly 2/(d+2) for every d≥ 2 . In the case of d=2 , this meets the previously well-known upper bound 1/2 tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than 1/2. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for d≥ 3 , the facet hypergraph ℱ() of a d -dimensional simplicial sphere has the chromatic number χ (ℱ())∈ O (n^(⌈ d/2⌉ -1)/d ) , where n is the number of vertices of . This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.