High Girth Steiner Triple Systems

Abstract

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 existence of designs by Glock, Kühn, 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 tuples of triples. In this talk we will also give a high-level overview of iterative absorption.

This work is joint with Matthew Kwan, Ashwin Sah, and Michael Simkin.

This is a seminar of the Combinatorial Mathematics Society of Australasia.

Direct Zoom link.

Meeting ID: 997 4063 1801 

School Seminar Series: 

• Combinatorics Seminar