Kahn–Kalai conjecture

The Kahn–Kalai conjecture, also known as the expectation threshold conjecture or more recently the Park-Pham Theorem, was a conjecture in the field of graph theory and statistical mechanics, proposed by Jeff Kahn and Gil Kalai in 2006. It was proven in a paper published in 2024.

Background
This conjecture concerns the general problem of estimating when phase transitions occur in systems. For example, in a random network with $$N$$ nodes, where each edge is included with probability $$p$$, it is unlikely for the graph to contain a Hamiltonian cycle if $$p$$ is less than a threshold value $$(\log N)/N$$, but highly likely if $$p$$ exceeds that threshold.

Threshold values are often difficult to calculate, but a lower bound for the threshold, the "expectation threshold", is generally easier to calculate. The Kahn–Kalai conjecture is that the two values are generally close together in a precisely defined way, namely that there is a universal constant $$K$$ for which the ratio between the two is less than $$K \log{\ell(\mathcal{F})}$$ where $$\ell(\mathcal{F})$$ is the size of a largest minimal element of an increasing family $$\mathcal{F}$$ of subsets of a power set.

Proof
Jinyoung Park and Huy Tuan Pham announced a proof of the conjecture in 2022; it was published in 2024.