Seminar

Zoom Link

In this talk, I will discuss several questions related to graph decompositions in random settings.  For example, how many edge-disjoint triangles can be packed into G(n,p)? For which d can the edges of a random n-vertex d-regular graph be decomposed into triangles?  What is the threshold at which the edges of K_n can be decomposed into triangles, when triangles are made available independently with probability p?

A decomposition of the edges of K_n into triangles can be viewed as a Steiner triple system, so these questions are closely related to Design Theory.  The development of the absorption method has led to asymptotic solutions to a number of longstanding problems in Design Theory, notably Keevash's 2014 proof of the Existence Conjecture for Combinatorial Designs as well as the iterative-absorption proof of Glock, Kühn, Lo, and Osthus.  Recently, Delcourt and Postle introduced the method of refined absorption and provided a third proof of the Existence Conjecture.  Although the previous approaches to the Existence Conjecture do not fare well in the random setting, I will explain how refined absorption can be used to make progress on these types of questions.  Joint work with Michelle Delcourt and Luke Postl

Graph decompositions in random settings via refined absorption