# Jinyoung Park and Huy Tuan Pham Prove the Kahn-Kalai Conjecture

Stanford mathematicians Jinyoung Park, a Szegö Assistant Professor, and Huy Tuan Pham, a PhD candidate, proved the Kahn-Kalai Conjecture: a major open problem in probabilistic combinatorics. Estimating the (probability) threshold for when a random graph (or much more generally, a random Boolean function) has a desired property is a fundamental and typically quite challenging problem. The expectation threshold, which is often easier to estimate, gives a lower bound on the threshold. Jeff Kahn and Gil Kalai conjectured in 2006 that the threshold and expectation threshold for any monotone property are always close (within a logarithmic factor of each other).

A weaker, fractional version of the conjecture due to Michel Talagrand was proved in a paper by Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Jinyoung Park that appeared last year in the Annals of Mathematics. This weaker version was already sufficient to give short proofs of various hard results and conjectures on thresholds.

The short preprint of Park and Pham proving the Kahn-Kalai conjecture can be found at this link.

Further information and a link to an IAS video about Dr. Park may be found at this link.

A Quanta article by Jordana Cepelwicz on this discovery can be found at this link.