12 problems

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…

Consider a 3-edge-colouring of the complete bipartite graph $K_{n,n}$. A partition into monochromatic paths is a partition of all vertices into vertex-disjoint paths, each monochro…

Bipartite upper-density conjecture. For every $k\in\mathbb{N}$ and every $K_{V,W}\in K_{\mathbb{N},\mathbb{N}}$, every $k$-edge-colouring of $K_{V,W}$ contains a monochromatic path…

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

Monochromatic path-length conjecture. Every edge-colouring of $K_n$ contains a monochromatic path of length

Let $k$ be a positive integer. A $k$-edge-coloured tournament is a tournament whose edges are assigned colours from a set of $k$ colours. A cycle of length $3$ is polychromatic whe…

A 3-edge-coloured tournament is a tournament whose edges are assigned one of three colours. A directed cycle is a directed dicycle, and a dicycle is polychromatic when its edges do…

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 $T$ be a random tournament on $n$ vertices, and let $m(T)$ denote the maximum number of vertices in a monochromatic path in a $2$-colouring of the edges of $T$, minimized over…

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 $G(n,p)$ be a random graph whose average degree tends to infinity, and consider arbitrary 2-colourings of its edges. Longest monochromatic path conjecture. The maximum length o…