Kahn–Kalai conjecture

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.[3]

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 ${\displaystyle K\log {\ell ({\mathcal {F}})}}$ where ${\displaystyle \ell ({\mathcal {F}})}$ is the size of a largest minimal element of an increasing family ${\displaystyle {\mathcal {F}}}$ of subsets of a power set.[4]

In 2022, Jinyoung Park and Huy Tuan Pham released a preprint containing a proposed proof of the conjecture.[3][4] The proof has been praised for its elegance and conciseness.[5]

• Percolation theory