Determine probability-threshold asymptotics from the expectation threshold

Determine the asymptotic behavior of the probability threshold p_c(\mathcal{Q}) for an arbitrary increasing graph property \mathcal{Q}, given its expectation threshold p_e(\mathcal{Q}), including whether such asymptotics can be determined even up to a constant factor.

Background

The paper studies the relationship between the probability threshold p_c(\mathcal{Q}) and the expectation threshold p_e(\mathcal{Q}) for increasing graph properties. The Park–Pham theorem gives the general upper bound p_c(\mathcal{Q})=O(p_e(\mathcal{Q})\log n_{\mathcal{Q}}), but the authors emphasize that the general asymptotic behavior of p_c remains unresolved.

For properties generated by copies of regular spanning graphs, the paper proves sharp asymptotics under expansion and automorphism conditions, while noting that graphs with many small subgraphs having small edge boundaries remain substantially more difficult. The stated problem asks for a general determination beyond the classes treated in the paper.