 
Summary: A note on regular Ramsey graphs
Noga Alon
Sonny BenShimon
Michael Krivelevich
July 16, 2009
Abstract
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.
1 Introduction
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 Ramseytype
problem was settled asymptotically by Kim [6] in 1995, after a long line of research; Kim showed that
n = (t2
/ log t). Recently, Bohman [1] gave an alternative proof of Kim's result by analyzing the socalled
trianglefree process, as proposed by Erdos, Suen and Winkler [3], which is a natural way of generating a
trianglefree 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 gadgetlike construction that transforms a trianglefree graph with no independent set of
