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



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> 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 a *finite* $\overline E\supseteq E$ and an $\overline S\supseteq S$ such that $(\overline E,\overline S)$ is real Steiner triple system?
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Can a partial Steiner triple system be completed?
Yes, there is quite a bit of literature on embedding partial triple systems. If you don't impose any further restrictions you can ensure an $S'$ of size approximately twice the size of $S$. In "Embedding partial Steiner triple systems so that their automorphisms extend", P.J. Cameron proves a stronger result that a partial triple system can be embedded in a Steiner triple system in a "homogeneous" way (every automorphism of the partial triple system extends to an automorphism of the Steiner triple system), however $S'$ ends up being much larger (exponential in the size of $S$). The article contains references to the previous work on the embedding problem. It is available from the ["Design Theory Resource Server"](http://designtheory.org/library/preprints/).


