  Since 1967, there have been hundreds of papers on finite graph-Ramsey theory
with many of the most important results being motivated by a series of
conjectures of Burr and Erd\H{o}s about graphs with linear Ramsey numbers. In a
sense, this paper begins a systematic study of (quantitative) countably
infinite graph-Ramsey theory, focusing on infinite analogues of the
Burr-Erd\H{o}s conjectures. The following are some of our main results.
  (i) Let $G$ be a countably infinite, (one-way) locally finite graph with
chromatic number $\chi<\infty$. Every 2-colored $K_\mathbb{N}$ contains a
monochromatic copy of $G$ with upper density at least $\frac{1}{2(\chi-1)}$.
  (ii) Let $G$ be a countably infinite graph having the property that there
exists a finite set $X\subseteq V(G)$ such that $G-X$ has no finite dominating
set (in particular, graphs with bounded degeneracy have this property, as does
the infinite random graph). Every finitely-colored $K_\mathbb{N}$ contains a
monochromatic copy of $G$ with positive upper density.
  (iii) Let $T$ be a countably infinite forest. Every 2-colored $K_\mathbb{N}$
contains a monochromatic copy of $T$ of upper density at least $1/2$. In
particular, this is best possible for $T_\infty$, the tree in which every
vertex has infinite degree.
  (iv) Surprisingly, there exist connected graphs $G$ such that every 2-colored
$K_\mathbb{N}$ contains a monochromatic copy of $G$ which covers all but
finitely many vertices of $\mathbb{N}$. In fact, we classify all forests with
this property.
