  Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive
integers. Answering a question of DeBiasio and McKenney, we construct a
$2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic
path has density~$0$.
  However, if we restrict the length of monochromatic paths in one colour, then
no example as above can exist: We show that every $(r+1)$-edge-coloured
complete symmetric digraph (of arbitrary infinite cardinality) containing no
directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered
by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochromatic complete symmetric
digraphs in colour $r+1$.
  Furthermore, we present a stability version for the countable case of the
latter result: We prove that the edge-colouring is uniquely determined on a
large subgraph, as soon as the upper density of monochromatic paths in colour
$r+1$ is bounded by $\prod_{i\in [r]}\frac{1}{\ell_i}$.
