Density of monochromatic infinite paths

For any subset $A subseteq mathbb{N}$, we define its upper density to be $limsup_{ n rightarrow infty } |A cap { 1, dotsc, n }| / n$. 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 ErdH{o}s and Galvin, and of DeBiasio and McKenney.