A triangle free configuration

Triangle-free subgraphs in the triangle-free process

Consider the triangle-free process, which is defined as follows. Start with G(0), an empty graph on n vertices. Given G(i − 1), let G(i) = G(i − 1) ∪ {g(i)}, where g(i) is an edge that is chosen uniformly at random from the set of edges that are not in G(i − 1) and can be added to G(i − 1) without creating a triangle. The process ends once a maximal triangle-free graph has been created. Let H b...

Triangle-free subgraphs at the triangle-free process

We consider the triangle-free process: given an integer n, start by taking a uniformly random ordering of the edges of the complete n-vertex graph Kn. Then, traverse the ordered edges 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 have been traversed for any fixed ε ∈ (0, ...

Free Triangle Orders

We define a new class of ordered sets, called free triangle orders. These are ordered sets represented by a left-to-right ordering on geometric objects contained in a horizontal strip in the plane. The objects are called “free triangles,” and have one vertex on each of the two boundaries of the strip and one vertex in its interior. These ordered sets generalize the classes of trapezoid and tria...

Measuring triangle-free graphs

The Triangle-Free Process

Consider the following stochastic graph process. We begin with G0, the empty graph on n vertices, and form Gi by adding a randomly chosen edge ei to Gi−1 where ei is chosen uniformly at random from the collection of pairs of vertices that neither appear as edges in Gi−1 nor form triangles when added as edges to Gi−1. Let the random variable M be the number of edges in the maximal triangle free ...