Upper density of monochromatic paths in edge-coloured infinite complete graphs and bipartite graphs

The upper density of an infinite graph $G$ with $V(G) \subseteq \mathbb{N}$ is defined as $\overline{d}(G) = \limsup_{n \rightarrow \infty}{|V(G) \cap \{1,\ldots,n\}|}/{n}$. Let $K_{\mathbb{N}}$ be the infinite complete graph with vertex set $\mathbb{N}$. Corsten, DeBiasio, Lamaison and Lang showed that in every $2$-edge-colouring of $K_{\mathbb{N}}$, there exists a monochromatic path with upper density at least $(12 + \sqrt{8})/17$, which is best possible. In this paper, we extend this result to $k$-edge-colouring of $K_{\mathbb{N}}$ for $k \ge 3$. We conjecture that every $k$-edge-coloured $K_{\mathbb{N}}$ contains a monochromatic path with upper density at least $1/(k-1)$, which is best possible (when $k-1$ is a prime power). We prove that this is true when $k = 3$ and asymptotically when $k =4$. Furthermore, we show that this problem can be deduced from its bipartite variant, which is of independent interest.