This is probably well-known... but I am afraid the literature on this subject bewilders me a little bit:

Suppose we have a

partial Steiner triple system, whereby I mean a finite set $E$ and a set $S$ of $3$-element subsets of $E$ such that every pair of elements in $E$ is in at most one element of $S$. Can one always find afinite$\overline E\supseteq E$ and an $\overline S\supseteq S$ such that $(\overline E,\overline S)$ is real Steiner triple system?