Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, May 29, 2012, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Luca Gugelmann

Sharp thresholds and hypergraph 2-colorability

A sharp threshold p0 for e.g. a monotone graph property A in the random graph G(n,p) is a function of n and A such that for any epsilon > 0 we have that for p < (1-epsilon)p0 the probability that A holds is asymptotically 0 while for (1+epsilon)p0 it is asymptotically 1. A threshold function which is not sharp is called coarse. Many interesting graph properties are conjectured to have sharp thresholds, however proving that this is the case (let alone determining p0 exactly) is pretty hard in general. In his PhD thesis Ehud Friedgut proved several results characterizing properties with coarse thresholds. These results then can be used to prove by contradiction that other thresholds must be sharp. In this talk I will give an overview of these results and present a nice proof by Friedgut and Alon showing that the threshold for 2-colorability of a k-uniform hypergraph (with k > 2) is sharp.

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