High-Girth Steiner Triple Systems

Combinatorics Seminar
Friday, December 2, 2022 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 202
Ashwin Sah – Massachusetts Institute of Technology – asah@mit.eduhttps://www.mit.edu/~asah/
Tom Kelly

We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth. Our proof builds on the method of iterative absorption for the existence of designs by Glock, Kü​hn, Lo, and Osthus while incorporating a "high girth triangle removal process". In particular, we develop techniques to handle triangle-decompositions of polynomially sparse graphs, construct efficient high girth absorbers for Steiner triple systems, and introduce a moments technique to understand the probability our random process includes certain configurations of triples.

(Joint with Matthew Kwan, Mehtaab Sawhney, and Michael Simkin) ​