Random graphs and thresholds

I will briefly introduce the notion of random graphs and some of their basic properties, mostly focusing on thresholds for increasing properties. I will also introduce "Kahn-Kalai expectation threshold conjecture" and explain the motivation behind it with some examples. If time permits, we will talk about a recent result of Jeff Kahn, Bhargav Narayanan, and myself stating that the threshold for the random graph G(n,p) to contain the square of a Hamilton cycle is 1/sqrt n, resolving a conjecture of Kühn and Osthus from 2012. The proof idea is motivated by the recent work of Frankston and the three aforementioned authors on a conjecture of Talagrand -- "a fractional version" of the Kahn-Kalai conjecture.