  For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite
monochromatic copy of $G$ in any $r$-coloring of the edges of the countably
infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is
natural to wonder how "large" of a monochromatic copy of $G$ we can find with
respect to some measure -- for instance, the density (or upper density) of the
vertex set of $G$ in the positive integers. Unlike finite Ramsey theory, where
this question has been studied extensively, the analogous problem for infinite
graphs has been mostly overlooked.
  In one of the few results in the area, Erd\H{o}s and Galvin proved that in
every 2-coloring 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 for some sequence $\epsilon_n\to 0$,
there exists a monochromatic path $P$ such that for infinitely many $n$, the
set $\{1,2,...,n\}$ contains the first $(\frac{1}{3+\sqrt{3}}-\epsilon_n)n$
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 $2/3$. We also consider related
problems for directed paths, trees (connected subgraphs), and a more general
result which includes locally finite graphs for instance.
