# Combinatorics and Discrete Probability Seminar: Notes on two-point concentration of the independence number of the random graph, by Tom Bohman

November 11, 2024

3:00 PM - 3:50 PM

Tom Bohman (Carnegie Mellon University): Notes on two-point concentration of the independence number of the random graph

It is well known that for any constant

probability p there exists a function k(n) such that the independence number of the binomial random graph G(n,p) is concentrated on two values (i.e.the independence number of G(n,p) is k(n) or

k(n)+1 with high probability). In this

talk we discuss the extension of this

result to p(n) that tends to 0 with n. In particular, we determine the probability at which two point concentration of the independence number of G(n,p) breaks down. We also discuss the independence number of G(n,m), and show that there is a range of values for m in which the independence number of G(n,m) is concentrated on two values while the independence number of the corresponding

G(n,p) is not concentrated on two values.

Joint work with Jakob Hofstad.

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## Date posted

Nov 4, 2024

## Date updated

Nov 4, 2024