For a tree $T$ whose bipartition classes have sizes $t_1 \ge t_2$, two simple constructions shows that the Ramsey number of $T$ is at least $\max\{t_1+2t_2,2t_1\}-1$. In 1974, Burr conjectured that equality holds for every tree. It turns out that Burr’s conjecture is false for certain trees called the double stars, though all of the known counterexamples have large maximum degrees. In 2002, Haxell, Łuczak, and Tingley showed that Burr’s conjecture is approximately true if one imposes a maximum degree condition.

We show that Burr’s conjecture holds for all trees with up to small linear maximum degrees. That is, there exists $c>0$ such that for every $n$-vertex tree $T$ with maximum degree at most $cn$ and bipartition class sizes $t_1\ge t_2$, its Ramsey number $R(T)$ is exactly $\max\{t_1+2t_2,2t_1\}-1$. We also generalise this result to determine the exact asymmetric Ramsey number $R(T,S)$ of two trees $T$ and $S$ under certain additional conditions, and construct examples showing that these conditions are necessary. 

This talk is based on joint work with Richard Montgomery and Matías Pavez-Signé.

First introduced by Fomin and Zelevinsky, cluster algebras are commutative rings that have many combinatorial properties. They have had many applications to both mathematics and physics. In this talk, I will first introduce cluster algebras and explore some of their properties. I will then move on to their applications, starting with dilogarithm identities and then moving to integrable systems and the thermodynamic Bethe ansatz (TBA). Time permitting, I will connect some of these ideas to the ODE/IM correspondence.