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Small Embeddings of Partial Steiner Triple Systems

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Abstract

It was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible math formula. This result is best possible in the sense that, for each math formula, there exists a partial Steiner triple system of order u that does not have an embedding of order v for any math formula. Many partial Steiner triple systems do have embeddings of orders smaller than math formula, but much less is known about when these embeddings exist. In this paper, we detail a method for constructing such embeddings. We use this method to show that each member of a wide class of partial Steiner triple systems has an embedding of order v for at least half (or nearly half) of the orders math formula for which an embedding could exist. For some members of this class we are able to completely determine the set of all orders for which the member has an embedding.

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