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A note on regular Ramsey graphs Sonny Ben-Shimon
 

Summary: A note on regular Ramsey graphs
Noga Alon
Sonny Ben-Shimon
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 Ramsey-type
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 so-called
triangle-free process, as proposed by Erdos, Suen and Winkler [3], 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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University
Krivelevich, Michael - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics