Embedding partial Steiner triple systems so that their automorphisms extend

Embedding partial Steiner triple systems so that their automorphisms extend

It is shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner triple system V on v points, in such a way that all automorphisms of U can be extended to V , for every admissible v satisfying v > g(u). We find exponential upper and lower bounds for g.

Embedding Partial Steiner Triple Systems

We prove that a partial Steiner triple system 8 of order n can be embedded in a Steiner triple system T of any given admissible order greater than 4w. Furthermore, if G(S), the missing-edge graph of S, has the property that A(G)<ri(n + l)l and \E(G)\ then # can be embedded in a Steiner triple system of order 2n +1, provided that 2w +1 is admissible. We also prove that if there is a partial Stei...

Automorphisms of Steiner triple systems

Abstract: Steiner triple systems are among the simplest and most intensively studied combinatorial designs. Their origins go back to the 1840s, and there exists by now a sizeable literature on the topic. In 1980, Babai proved that almost all Steiner triple systems have no nontrivial automorphism. On the other hand, there exist Steiner triple systems with large automorphism groups. We will discu...

The embedding problem for partial Steiner triple systems

The system has the nice property that any pair of distinct elements of V occurs in exactly one of the subsets. This makes it an example of a Steiner triple system. Steiner triple systems first appeared in the mathematical literature in the mid-nineteenth century but the concept must surely have been thought of long before then. An excellent historical introduction appears in [7]. As pointed out...

Embedding Steiner triple systems in hexagon triple systems