| Abstract: | In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u < v < 2u + 1, we ask for the minimum r such that there exists a Steiner triple system  such that some partial system  can be completed to an STS , where |?| = r. In other words, in order to “quasi‐embed” an STS(u) into an STS(v), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity (u(u ? 1)/6) ? r as the maximum intersection of an STS(u) and an STS(v) with u < v. We conjecture that the necessary minimum r = (v ? u) (2u + 1 ? v)/6 can be achieved, except when u = 6t + 1 and v = 6t + 3, in which case it is r = 3t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v ? u = 2 and 4, asymptotically solve the cases v ? u = 6, 8, and 10, and further show for given v ? u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v ? u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc. |
