4-cycles at the triangle-free process

We consider the triangle-free process: given an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph K n . Then, traverse the edges of K n according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving graph - unless its addition creates a triangle. We study the evolving graph at around the time where Θ ( n 3 / 2 + ϵ ) edges of K n have been traversed for any fixed ϵ ∈ ( 0 , 10 − 10 ) . At that time, we give a tight concentration result for the number of copies of the 4-cycle in the evolving graph. Our analysis combines Spencerʹs original branching process approach for analysing the triangle-free process with the semi-random method.