Pokrovskiy's bounded-defect monochromatic cycle conjecture

Let $r$ be a positive integer, and let a complete graph have its edges coloured with $r$ colours. A collection of vertex-disjoint monochromatic cycles consists of cycles with pairwise disjoint vertex sets, each cycle having edges of a single colour. Pokrovskiy's conjecture. There is a constant $c_r$, depending only on $r$, such that every $r$-edge-coloured complete graph has $r$ vertex-disjoint monochromatic cycles covering all but at most $c_r$ vertices. This is presented as a widely open alternative to the refuted Erdős–Gyárfás–Pyber conjecture; the best general result cited in the source uses at most $100r\log r$ monochromatic cycles for sufficiently large graphs.