The Asymptotic Lower Bound on the Diagonal Ramsey Numbers: A Closer Look

Abstract

The classical diagonal Ramsey number R(k, k) is defined, as usual, to be the smallest integer n such that any two-coloring of the edges of the complete graph Kn on n vertices yields a monochromatic k-clique. It is well-known that R(3, 3) = 6 and R(4, 4) = 18; the values of R(k, k) for k ⩾ 5, are, however, unknown. The Lovász local lemma has been used to provide the best-known asymptotic lower bound on R(k, k), namely $\frac{{\sqrt 2 }}{e}k2^{k/2} (1 + ))(k \to \infty )$. In other words, if one randomly two-colors the edges of Kn, then P(X ₀ = 0) > 0 if

$$n \leqslant \frac{{\sqrt {2} }}{e}k{{2}^{{k/2}}}(1 + o(1)), $$

where X0 denotes the number of monochromatic k-cliques. We use univariate and process versions of the Stein-Chen technique to show that much more is true under condition (♣): in particular, we prove (i) that the distribution of X0 can be approximated by that of a Poisson random variable, and that (ii) with Xj representing the number of k-cliques with exactly j edges of one color, the joint distribution of (X0, X1,..., Xb) can be approximated by a multidimensional Poisson vector with independent components provided that b = o(k/log k).

This research was partially supported by NSF Grant DMS-9200409. Part of the work reported herein was conducted at the Institüt für Angewandte Mathematik, Universität Zürich. Useful discussions with Andrew Barbour are gratefully acknowlegde, as is the generous travel support derived from a Schweizerischer Nationalfonds Grant.