The Kahn–Kalai conjecture, also known as the expectation threshold conjecture, is a conjecture in the field of graph theory and statistical mechanics, proposed by Jeff Kahn and Gil Kalai in 2006.^[1][2]

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.^[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]

Proof

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]

References

See also

• Percolation theory