Gyarfas's conjecture on covering edge-colored complete graphs by r-1 monochromatic trees

For every sufficiently large prime $q$ (September 27 paper), and for $r=s^n+1$ with $s\equiv2\pmod3$ a large prime and $n$ large odd (September 23 paper), there is a finite complete graph with an $r$-edge-coloring whose vertices need exactly $r$ monochromatic trees to cover them; the same holds for connected monochromatic subgraphs. Built from the Ryser counterexamples by coloring each pair of hyperedges by a part where they meet. Not addressed: small $r$, or the distinct vertex-disjoint tree-partition conjecture of Erdos-Gyarfas-Pyber.

For every sufficiently large prime $q$ (September 27 paper), and for $r=s^n+1$ with $s\equiv2\pmod3$ a large prime and $n$ large odd (September 23 paper), there is a finite complete graph with an $r$-edge-coloring whose vertices need exactly $r$ monochromatic trees to cover them; the same holds for connected monochromatic subgraphs. Built from the Ryser counterexamples by coloring each pair of hyperedges by a part where they meet. Not addressed: small $r$, or the distinct vertex-disjoint tree-partition conjecture of Erdos-Gyarfas-Pyber.