5 problems

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

Let $k$ be a positive integer. A $k$-edge-coloured finite tournament is a finite tournament whose edges are assigned one of $k$ colours, and a set $S\subseteq V(T)$ is reachable fr…

Let $H$ be a graph with chromatic number $k+1$, and let $G$ be an $t$-edge-colored graph on $n$ vertices such that $H$ is not a subgraph of the complement $\overline{G}$. H-free-co…

Infinite bipartite monochromatic path conjecture. The vertices can be partitioned into $2r-1$ disjoint monochromatic paths.

Let $K_{n,n}$ be a balanced complete bipartite graph whose edges are coloured with $r$ colours. A vertex-partition into monochromatic paths is a partition of all vertices into path…