  A famous theorem of Kirkman says that there exists a Steiner triple system of
order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erd\H{o}s conjectured
that one can find so-called `sparse' Steiner triple systems. Roughly speaking,
the aim is to have at most $j-3$ triples on every set of $j$ points, which
would be best possible. (Triple systems with this sparseness property are also
referred to as having high girth.) We prove this conjecture asymptotically by
analysing a natural generalization of the triangle removal process. Our result
also solves a problem posed by Lefmann, Phelps and R\"odl as well as Ellis and
Linial in a strong form, and answers a question of Krivelevich, Kwan, Loh, and
Sudakov. Moreover, we pose a conjecture which would generalize the Erd\H{o}s
conjecture to Steiner systems with arbitrary parameters and provide some
evidence for this.
