Density of monochromatic infinite paths

For any subset $A\subseteq \mathbb{N}$, we define its upper density to be ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}\mathrm{\mid }A\cap \{1,\dots ,n\}\mathrm{\mid }\mathrm{/}n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erd\H{o}s and Galvin, and of DeBiasio and McKenney.