5-sparse Steiner Triple Systems of Order $n$ Exist for Almost All Admissible $n$

Abstract

Steiner triple systems are known to exist for orders $n \equiv 1,3$ mod $6$, the admissible orders. There are many known constructions for infinite classes of Steiner triple systems. However, Steiner triple systems that lack prescribed configurations are harder to find. This paper gives a proof that the spectrum of orders of 5-sparse Steiner triple systems has arithmetic density $1$ as compared to the admissible orders.