Subgraph Kahn–Kalai conjecture for growing graphs

Prove the subgraph version of the Kahn–Kalai conjecture for growing graph sequences H=H_n, including weakly dense graph sequences, and determine whether the subgraph expectation threshold characterizes the critical threshold without the presently unresolved limitations.

Background

The paper relates the MMSE transition thresholds in the planted subgraph model to modified subgraph expectation thresholds. The authors explain that the probabilistic intuition behind this relationship relies on the Kahn–Kalai conjecture, which concerns the critical probability at which a graph appears in an Erdős–Rényi random graph.

Although the paper proves the MMSE characterization directly, it does not prove the underlying subgraph Kahn–Kalai conjecture. The unresolved case is specifically a growing graph H=H_n, even under the weak-density assumption used for the paper’s main planted-subgraph results; the authors also note that the conjecture is expected to provide only logarithmic-factor accuracy in its usual formulation.

References

Perhaps the most crucial issue is that the subgraph version of the Kahn--Kalai conjecture remains open for a growing $H=H_n$ (even restricted to weakly dense $H$); moreover, even if proved, it is only expected to approximate $p_c(H)$ to within a multiplicative $O(\log |H|)$ factor of the true critical threshold.

— The Fundamental Limits of Recovering Planted Subgraphs  (2503.15723 - Lee et al., 19 Mar 2025) in Section 1, Introduction, paragraph beginning “However, this line of argument cannot be applied directly”

Dubroff, Kahn, and Park proposed the following counting conjecture, which we state here as an equivalent spread bound. There is an absolute constant C such that ν_H is Cq_H-spread for every nonempty H⊆K_n.

— Fractional expectation thresholds and the "second" Kahn-Kalai conjecture  (2609.20546 - Tran, 17 Sep 2026) in Section 1, immediately following Theorem 1.3 (Conjecture D)