Abstract

Kahn and Kalai conjectured that the threshold of an increasing property is always within a logarithmic factor of the expectation threshold, a quantity often much easier to compute. The Kahn-Kalai conjecture directly implies a number of difficult results in probabilistic combinatorics. I will discuss recent joint work with Jinyoung Park that resolves the Kahn-Kalai conjecture. Time permitting, I will discuss our resolution of some conjectures and questions of Talagrand, which were in fact the precursor to the resolution of the Kahn-Kalai conjecture.

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