The potential to improve the choice: List coloring for geometric hypergraphs

conflict-free if for every hyperedge S in E there is at least one
vertex whose color is distinct from the colors of all other vertices
in S. This notion has been studied for several hypergraphs and in
various generalizations.
We study the list version of this notion: In this version we are
interested in the minimum number k=k(H) such that if each vertex v is
associated with a set of colors L_v of size k then one can pick a
color for each vertex v from its "list" L_v such that the resulting
coloring is conflict-free. Denote this number by ch(H) (the conflict
free "choice"number of H). It is easy to see that the minimum number
of colors needed for a conflict-free coloring of H is bounded by
ch(H).
Let C be some absolute constant. We prove that for hypergraphs H with
n vertices which are hereditarily C colorable (in the
non-monochromatic sense) we have ch(H) = O(\log n) and this bound is
asymptotically tight. Moreover, we show that one can color the
vertices of such a hypergraph from lists of size O(\log n) such that
the maximum color in any hyperedge is unique. The proof is
constructive and uses a suitable potential function for constructing
such coloring and analyzing the size of lists needed.

Joint work with Panagiotis Cheilaris and Marek Sulovsky