Combinatorics Seminar
When: Sunday, May 6, 10am
Where: Schreiber 309
Speaker: Amit Weinstein, Tel Aviv University
Title: Semi-Strong Coloring of Intersecting Hypergraphs
Abstract:
For any c >= 2, a c-strong coloring of a hypergraph G is an assignment of
colors to the vertices of G such that for every edge e of G, the vertices
of e are colored by at least min{c,|e|} distinct colors. The hypergraph
G is t-intersecting if every two edges of G have at least t vertices
in common. We ask: for fixed c >= 2 and t >= 1, what is the minimum
number of colors that is sufficient to c-strong color any t-intersecting
hypergraph ?
The purpose of this talk is to answer the question for some values of
t and c and, more importantly, to describe the settings for which the
question is still open. We show that when t <= c-2, no finite number of
colors is sufficient to c-strong color all t-intersecting hypergraphs. It
is still unknown whether a finite number of colors suffices for the same
task when t = c-1 and c > 2. In the last case, when t >= c, we show with
a probabilistic argument that a finite number of colors is sufficient
to c-strong color all t-intersecting hypergraphs, but a large gap still
remains between the best upper and lower bounds on this number.
Joint work with Eric Blais and Yuichi Yoshida