Louis DeBiasio (Miami University) Infinite graphRamsey theory Abstract: Ramsey's theorem guarantees a monochromatic copy of any countably infinite graph $G$ in any $r$coloring of the edges of the complete graph $K_\mathbb{N}$. It is natural to wonder how "large" of a monochromatic copy of $G$ we can find with respect to some measure  for instance, the (upper) density of the vertex set of $G$ in $\mathbb{N}$. Unlike finite graphRamsey theory, where this question has been studied extensively, the infinite version has been mostly overlooked. Erdős and Galvin proved that in every 2coloring of $K_\mathbb{N}$, there exists a monochromatic path whose vertex set has upper density at least $2/3$, but it is not possible to do better than $8/9$. They also showed that there exists a monochromatic path $P$ such that for infinitely many $n$, the set $\{1,2,...,n\}$ contains the first $\frac{n}{3+\sqrt{3}}$ vertices of $P$, but it is not possible to do better than $2n/3$. We improve both results, in the former case achieving an upper density at least $3/4$ and in the latter case obtaining a tight bound of $2/3$. Inspired by this, we consider infinite analogs of wellknown finite results on directed paths, trees (connected subgraphs), and graphs of bounded maximum degree/chromatic number. Joint work with Paul McKenney 
