Summary: A note on regular Ramsey graphs
July 16, 2009
We prove that there is an absolute constant C > 0 so that for every natural n there exists a triangle-
free regular graph with no independent set of size at least C
n log n.
A major problem in extremal combinatorics asks to determine the maximal n for which there exists a graph
G on n vertices such that G contains no triangles and no independent set of size t. This Ramsey-type
problem was settled asymptotically by Kim  in 1995, after a long line of research; Kim showed that
n = (t2
/ log t). Recently, Bohman  gave an alternative proof of Kim's result by analyzing the so-called
triangle-free process, as proposed by Erdos, Suen and Winkler , which is a natural way of generating a
triangle-free graph. Consider now the above problem with the additional constraint that G must be regular.
In this short note we show that the same asymptotic results hold up to constant factors. The main ingredient
of the proof is a gadget-like construction that transforms a triangle-free graph with no independent set of