Location: Levy 6 hall

zoom link: https://huji.zoom.us/j/85722726344?pwd=N0paMnc1dUlVdytIcjZBS0k5MVNiQT09
Meeting ID: 857 2272 6344
Passcode:  711270

Title: Transversals and colorings of simplicial spheres

Abstract: Motivated from the surrounding property of a point set in \RR^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 \frac{2}{d+2} for every d\geq 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\geq 3, the facet hypergraph \mathcal{F}(\sK) of a d-dimensional simplicial sphere \sK
has the chromatic number \chi(\mathcal{F}(\sK)) \in O(n^{\frac{\lceil d/2\rceil-1}{d}}), where n is the number of vertices of \sK.
This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.