I will discuss our resolution of a 1973 conjecture of Erdős on the existence of Steiner triple systems with arbitrarily large girth. (A Steiner triple system (STS) is a collection of triples on n vertices such that each pair is contained in exactly one triple. Its girth is the smallest g > 3 for which there exist g vertices spanning at least g-2 triples.) Our construction builds on the methods of iterative absorption (Glock, Kühn, Lo, and Osthus) and the high-girth triangle removal process (Glock, Kühn, Lo, and Osthus; Bohman and Warnke).

I will begin by motivating the problem and giving an overview of iterative absorption. As time permits, I will discuss the difficulties present in the high-girth setting and some ideas needed to overcome these challenges, related to sparsification, efficient absorption, and retrospective analysis of random processes.

This is joint work with Matthew Kwan, Ashwin Sah, and Mehtaab Sawhney. Based on arXiv:2201.04554