Combinatorics Seminar

For positive integers $s$ and $k$, the Ramsey number $r(s,k)$ is the minimum integer $n$ such that any graph on $n$ vertices contains a clique of size $s$ or an independent set of size $k$. The main focus of this talk will be the asymptotic regime of so-called off-diagonal Ramsey numbers, where the parameter $s$ is fixed and $k$ tends to infinity. I will give an overview of the ideas that lead to recent advances on the asymptotic bounds for these numbers. Finally, I will sketch the proof of a recent result determining the exponent of $r(s, k)$ for any fixed $s$.