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.

See also

• Percolation theory