Determining Ramsey numbers using finite geometry

Sam Mattheus and Jacques Verstraete made a remarkable breakthrough for Ramsey numbers , and Anurag Bishnoi wrote a beautiful blog post about it. Congratulations Sam and Jacques! Mattheus and Verstraete show that .

Sam Mattheus and Jacques Verstraete have posted a preprint today where they solve the classic open problem of determining the asymptotics of the Ramsey number $latex r(4, t)$. They show that

$latex r(4, t) geq c frac{t^3}{log^4 t}$

which is just a factor of $latex log^2 t$ away from the upper bound. The only other off-diagonal Ramsey number for which we knew the correct asymptotics prior to their work was $latex r(3, t)$, and the best lower bounds on $latex r(4, t)$ were $latex c’ t^{5/2}/log^2 t$. These earlier bounds are in fact at the limit of what could be proved using the random $latex H$-free process. That barrier has finally been broken by using completely different techniques involving finite geometry! It’s an amazing breakthrough that builds up on the recent developments in Ramsey theory using finite geometry (see this for an online minicourse I gave in 2021…