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.
[3] This conjecture concerns the general problem of estimating when phase transitions occur in systems.
nodes, where each edge is included with probability
[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
is the size of a largest minimal element of an increasing family
[3] Jinyoung Park and Huy Tuan Pham announced a proof of the conjecture in 2022; it was published in 2024.