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.^[1][2] It was proven in a paper published in 2024.^[3]

Background

This conjecture concerns the general problem of estimating when phase transitions occur in systems.^[1] For example, in a random network with ${\displaystyle N}$ nodes, where each edge is included with probability ${\displaystyle p}$, it is unlikely for the graph to contain a Hamiltonian cycle if ${\displaystyle p}$ is less than a threshold value ${\displaystyle (\log N)/N}$, but highly likely if ${\displaystyle p}$ exceeds that threshold.^[4]

Threshold values are often difficult to calculate, but a lower bound for the threshold, the "expectation threshold", is generally easier to calculate.^[1] 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 ${\displaystyle K}$ for which the ratio between the two is less than ${\displaystyle K\log {l({\mathcal {F}})}}$ where ${\displaystyle l({\mathcal {F}})}$ is the size of a largest minimal element of an increasing family ${\displaystyle {\mathcal {F}}}$ of subsets of a power set.^[3]

Proof

Jinyoung Park and Huy Tuan Pham announced a proof of the conjecture in 2022; it was published in 2024.^[4][3]