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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 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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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".

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Thank you! ${}$ –  Mariano Suárez-Alvarez Aug 17 '11 at 6:07
    
Beat me by 2 minutes. –  Gerry Myerson Aug 17 '11 at 6:08

Have you seen Peter J Cameron, Embedding partial Steiner triple systems so that their automorphisms extend, available at this site? It says (among other things), "Lindner [4] conjectured that, more generally, a PSTS of order u can be embedded in an STS of any admissible order greater than 2u. The conjecture has recently been proved by Bryant [2], though this is not yet published." PSTS is partial Steiner triple system. The references are C. C. Lindner, A partial Steiner triple system of order $n$ can be embedded in a Steiner triple system of order $6n + 3$, J. Combinatorial Theory Ser. A 18 (1975), 349–351, and D. Bryant, talk at 19th ACCMCC, Taupo, New Zealand, December 2004.

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Thank you too! :) –  Mariano Suárez-Alvarez Aug 17 '11 at 6:10

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