The New Result on Off-diagonal Ramsey Numbers

(All references in this blog post can be found in the main article the post is about which is here.)Recall that \(R(a,b)\) is the least \(n\) so that, for all 2-colorings of the edges of \(K_n\), there is either a RED \(a\)-clique or a BLUE \(b\)-clique.\(R(k,k)\) has been well studied and is often called \(R(k)\).However, today we are concerned with \(R(a,k)\) \(a\) is fixed and \(k\) goes to infinity.1) In 1995 Jeong Han Kim showed \(R(3,k)\) is asy \(\Theta(\frac{k^2}{\log k})\). At the workshop In Ramsey Theory: Yesterday, Today, and Tomorrow, Edited by Alexander Soifer, 2011, i Joel Spencer give a great talk titled80 years of \(R(3,k)\).The title implies that the problem was open for 80 years, but 40 years is a better estimate.The general sense I got from both Joel and the audience is that \(R(4,k)\) is a much harder problem.2) In 2023 there was substantial progress on \(R(4,k)\). Sam Mattheus and Jacques Verstraete showed\(R(4,k) = \Omega(\frac{k^3}{\log^4 k})\)Combined with…