9 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 Kn. Then, traverse the edges of Kn 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) edges of Kn have been traversed for any fixed ε ∈ (0, 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 uses in part Spencer’s original branching process approach for analysing the triangle-free process, coupled with the semi-random method.