Geometric selection theorems
Geometric selection theorems

Boris Bukh, Princeton University and UCLA
Fine Hall 224
In combinatorial geometry one frequently wants to select a point or a set of points that meets many simplices of a given family. The two examples are choosing a point in many simplices spanned by points of some $P$ in $R^d$, and choosing a small set of points which meets the convex hull of every large subset of $P$ (the weak epsilonnet problem). I will present a new class of constructions that yield the first nontrivial lower bound on the weak epsilonnet problem, and improve the best bounds for several other selection problems. Joint work with Jiří Matoušek and Gabriel Nivasch.