Determine the general anti-Ramsey threshold for random graphs

Determine the anti-Ramsey threshold of the binomial random graph G_{n,p} for every fixed graph H, beyond the dense-graph regime established in the paper.

Background

For graphs G and H, G is anti-Ramsey for H if every proper edge-colouring of G contains a rainbow copy of H. The paper studies the threshold at which the binomial random graph G_{n,p} acquires this property. Prior work established a general upper bound at p of order n^{-1/m_2(H)}, but this bound is not always tight: for example, triangles and several other graphs become anti-Ramsey at substantially lower densities.

The paper proves that the threshold has order n^{-1/m_2(H)} for graphs H with m_2(H) at least 19. Thus, the broader problem of locating the anti-Ramsey threshold for arbitrary fixed graphs remains unresolved, particularly for sparse graphs and the known families of counterexamples to the m_2(H)-threshold prediction.