Abstract: We prove a 1973 conjecture of Erdős on the existence of Steiner triple systems with arbitrarily high girth. (The girth of an STS is the smallest g>3 for which there exist g vertices spanning at least g-2 triangles.) 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 and independently Bohman and Warnke). In particular, we extend iterative absorption to handle triangle-decompositions of sparse graphs. We also introduce a moments-based technique, similar to Kim--Vu polynomial concentration for dependent random variables, which allows a unified approach to control statistics of complex random processes.

The talk will include an overview of iterative absorption.

This is joint work with Matthew Kwan, Ashwin Sah, and Mehtaab Sawhney.

Based on arXiv:2201.04554