# Combinatorics Seminar

When: Sunday, February 16, 10am

Where: Schreiber 309

Speaker: Danny Vilenchik, Weizmann

Title: Chasing the k-colorability threshold

## Abstract:

In this talk I will present a substantially improved lower bound on the $k$-colorability threshold of the random graph $G(n,m)$ with $n$ vertices and $m$ edges. The new lower bound is 1.39 less than the $2k\ln k-\ln k$ first-moment upper bound (and 0.39 less than the 2k\ln k-\ln k-1 physics conjecture). By comparison, the best previous bounds left a gap of about 2+\ln k, unbounded in terms of the number of colors [Achlioptas, Naor: Annals of Mathematics 2005]. Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments. Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of $k$-colorings of the random graph.

Joint work with Amin Coja-Oghlan. Paper appeared in FOCS 2013.

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