  We prove that in every $2$-colouring of the edges of $K_\mathbb{N}$ there
exists a monochromatic infinite path $P$ such that $V(P)$ has upper density at
least ${(12+\sqrt{8})}/{17} \approx 0.87226$ and further show that this is best
possible. This settles a problem of Erd\H{o}s and Galvin.
