Georgia Institute of Technology College of Sciences

Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a quasikernel if $Q$ is an independent set and if every vertex of $D$ can be reached from $Q$ by a path of length at most 2.  The Small Quasikernel Conjecture of P.L.\ Erd\H{o}s and Sz\'ekely from 1976 states that every $n$-vertex source-free digraph $D$ contains a quasikernel of size at most $\frac{1}{2}n$.  Despite being posed nearly 50 years ago, very little is known about this conjecture, with the only non-trivial upper bound of $n-\frac{1}{4}\sqrt{n\log n}$ being proven recently by ourself.  We discuss this result together with a number of other related results and open problems around the Small Quasikernel Conjecture.

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}3\] for all sufficiently large $t$. 

Our proof employs many recent results on the chromatic number of locally sparse graphs. In particular, I will highlight a new result on the chromatic number of degenerate graphs, which leads to several intriguing open problems.

A Latin square is an nxn grid filled with n symbols such that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of n cells such that each row, column and symbol appears exactly once in the collection.

Latin squares were introduced by Euler in the 1700s and he was interested in the question of when a Latin square decomposes fully into transversals. Equivalently, when does a Latin square have an 'orthogonal mate'?

We'll discuss the history of this question, and some upcoming joint work with Richard Montgomery.

A hereditary class $\mathcal C$ of graphs is said to have the Erdős–Hajnal property if every $n$-vertex graph in $\mathcal C$ has a clique or stable set of size at least $n^c$. We discuss a proof of a conjecture of Chernikov–Starchenko–Thomas and Fox–Pach–Suk that for every $d\ge1$, the class of graphs of VC-dimension at most $d$ has the Erdős–Hajnal property. Joint work with Alex Scott and Paul Seymour.