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(K) of a d-dimensional simplicial sphere K has the chromatic number χ(F(K))∈O(n^{(⌈d/2⌉-1)/d}), where n is the number of vertices of K. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.