Large girth approximate Steiner triple systems

TL;DR

This paper proves the existence of large girth approximate Steiner triple systems with many triples, using a constrained random process that ensures high girth and near-quadratic size.

Contribution

It introduces a natural constrained random process that constructs approximate Steiner triple systems with arbitrarily high girth and near-quadratic size, answering a long-standing question.

Findings

Existence of approximate Steiner triple systems with high girth and many triples.

The constrained random process typically produces systems with (1/6-o(1))n^2 triples.

Girth larger than any fixed  is achievable with high probability.

Abstract

In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than \ell. Our result is best possible up to the o(1)-term, which is a negative power of n.