Bounded diameter monochromatic component covers
Combinatorics
2026-01-07 v2
Abstract
Ryser conjectured that every -edge-coloured complete graph can be covered by monochromatic trees. Motivated by a question of Austin in analysis, Mili\'cevi\'c predicted something stronger -- that every -edge-coloured complete graph can be covered by monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Mili\'cevi\'c's Conjecture, most notably that it is true for . We also obtain several new cases of a generalization of Mili\'cevi\'c's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats.
Cite
@article{arxiv.2507.05842,
title = {Bounded diameter monochromatic component covers},
author = {Alexey Pokrovskiy},
journal= {arXiv preprint arXiv:2507.05842},
year = {2026}
}