Summary: Hypergraph list coloring and Euclidean Ramsey Theory
April 28, 2010
A hypergraph is simple if it has no two edges sharing more than a single vertex. It is s-list
colorable (or s-choosable) if for any assignment of a list of s colors to each of its vertices, there is
a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic.
We prove that for every positive integer r, there is a function dr(s) such that no r-uniform simple
hypergraph with average degree at least dr(s) is s-list-colorable. This extends a similar result
for graphs, due to the first author, but does not give as good estimates of dr(s) as are known for
d2(s), since our proof only shows that for each fixed r 2, dr(s) 2crsr-1
. We use the result to
prove that for any finite set of points X in the plane, and for any finite integer s, one can assign
a list of s distinct colors to each point of the plane, so that any coloring of the plane that colors
each point by a color from its list contains a monochromatic isometric copy of X.
AMS Subject Classification: 05C15, 05C35, 05C65.
Keywords: Hypergraphs, list coloring, average degree, Euclidean Ramsey Theory.