The upper-density conjecture for monochromatic paths in infinite complete graphs

Let $K_{\mathbb{N}}$ be the infinite complete graph on the strictly positive integers, and let $\overline{d}(P)$ denote the upper density of the vertex set of a path $P$. A $k$-edge-colouring uses colours $1,2,\ldots,k$.

Upper-density conjecture. For every integer $k\geq 3$, every $k$-edge-colouring of $K_{\mathbb{N}}$ contains a monochromatic path $P$ such that

$$\overline{d}(P)\geq \frac{1}{k-1}.$$

The bound is conjectured to be best possible when $k-1$ is a prime power; the paper proves the claim for $k=3$ and asymptotically for $k=4$, while the general case remains open.