Seminar series
Date
Mon, 04 Feb 2008
13:30
13:30
Location
L3
Speaker
David Conlon
Organisation
Cambridge
Let d be a fixed natural number. There is a theorem, due to Chvátal, Rodl,
Szemerédi and Trotter (CRST), saying that the Ramsey number of any graph G
with maximum degree d and n vertices is at most c(d)n, that is it grows
linearly with the size of n. The original proof of this theorem uses the
regularity lemma and the resulting dependence of c on d is of tower-type.
This bound has been improved over the years to the stage where we are now
grappling with proving the correct dependency, believed to be an
exponential in d. Our first main result is a proof that this is indeed the
case if we assume additionally that G is bipartite, that is, for a
bipartite graph G with n vertices and maximum degree d, we have r(G)