Problem 811

Erdős Problem #811: declared status 'open'. Formalized: no. Suppose $n\equiv 1\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\lfloor n/m\rfloor$ many edges of each colours. For which graphs $G$ is it true that, if $m=e(G)$, for all large $n\equiv 1\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.) Current best: They conjecture that $K_m$ lacks this property for all $m\geq 4$. Prize: no. Tags: graph theory, ramsey theory.