There exists a universal constant C>0 such that, for every finite graph H, the fractional expectation threshold for containing H satisfies p_ E^(H)≤ C p_ E(H), where p_ E(H) is the expectation threshold defined by the condition that the expected number of copies of H is at least 1.

Second Kahn–Kalai conjecture

The Second Kahn–Kalai conjecture predicts that the threshold for containing a graph is controlled by its expectation threshold without extra logarithmic losses. The retrieved sources record substantial partial progress, but no result settling the conjecture for every graph.

Known results

The 2022 modified-threshold theorem proves $p_{\mathsf c}(H)\leq L\tilde p_{\mathsf E}(H)\log e(H)$ , but explicitly leaves the original conjecture open.
A 2025 result establishes $p_{\mathbb E}^{*}(H)<Kp_{\mathbb E}(H)\log^{2}n$ and consequently $p_{\mathsf c}(H)=O(p_{\mathbb E}(H)\log^{3}n)$ .
Another 2025 result proves the relevant fractional statement for cliques, cycles, trees of bounded maximum degree, and forests.

September 2026 progress

A September 17, 2026 report attributes to Tuan Tran a proof of the fractional expectation-threshold statement for the stated graph classes, including a logarithmic-loss bound in general. This is claimed progress rather than a resolution for all graphs.

Current status (as of September 2026): The conjecture has claimed proofs for substantial graph families and quantitative general bounds, but remains open for arbitrary graphs.

Second Kahn–Kalai conjecture

Primary source

Additional references

Progress summary

Solutions 0