Density of monochromatic infinite subgraphs II

In 1967, Gerencs\'er and Gy\'arf\'as proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$ there is a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erd\H os regarding the Ramsey numbers of trees, graphs with bounded maximum degree, and graphs with bounded degeneracy. In 1993, Erd\H os and Galvin \cite{EG} began the investigation of a countably infinite analogue of the Gerencs\'er and Gy\'arf\'as result: What is the largest $d$ such that in every $2$-coloring of $K_\mathbb{N}$ there is a monochromatic infinite path with upper density at least $d$. Erd\H os and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, it was finally shown that $d={(12+\sqrt{8})}/{17}$. 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. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.