Balanced $e(G)$-colourings of $K_n$: which graphs $G$ are forced to appear rainbow? (Erdős #811)

An edge-colouring of the complete graph $K_n$ with $m$ colours is called balanced if, whenever $n\equiv 1\pmod m$, every vertex is incident to exactly $\lfloor n/m\rfloor$ edges of each colour. For which graphs $G$ is the following true: setting $m=e(G)$ (the number of edges of $G$), for all sufficiently large $n\equiv 1\pmod m$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$ — a subgraph isomorphic to $G$ whose $e(G)$ edges all receive distinct colours? A specific challenge, for $m=6$: must every balanced $6$-colouring of $K_{6n+1}$ contain both a rainbow $C_6$ and a rainbow $K_4$ (each has $6$ edges)?

Acceptance. FULLY RESOLVES: a complete characterisation of the graphs $G$ for which every balanced $e(G)$-colouring of $K_n$ (with $n\equiv 1\pmod{e(G)}$, $n$ large) forces a rainbow $G$ — a proven necessary-and-sufficient condition — or, at minimum for the flagged $m=6$ challenge, a full proof deciding whether every balanced $6$-colouring of $K_{6n+1}$ contains a rainbow $C_6$ (the $K_4$ part is already known to fail via Clemen–Wagner, so a rederivation of it alone does not count). A machine-checkable proof (Lean) is preferred, otherwise a complete written proof. ADVANCES: (a) decide the property for a specific previously-open graph $G$ (e.g. prove $C_6$ is or is not forced), with proof; (b) improve the Erdős–Tuza bounds $\lfloor n/6\rfloor\leq d_{C_4}(n)\leq(1/4-c)n$ strictly beyond what the background states, with proof; or (c) for a new record $n$, exhaustively certify via SAT/ILP whether all balanced colourings of $K_n$ force a rainbow target $G$, extending the known constructions with an exhaustiveness certificate. Deliver the proof file, the improved-bound proof, or the search code plus certificate.