A proof of Lindner's conjecture on embeddings of partial Steiner triple systems

Abstract

Lindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if v ≡ 1, 3 (mod 6) and v ≥ 2u + 1 is proved.

Embedding partial odd‐cycle systems in systems with orders in all admissible congruence classes

“…It is shown in [2] that any partial 3-cycle system of order u can be embedded in a 3-cycle system of order v if v ≥ 2u +1, v is odd and is an m-cycle system of order v that contains an isomorphic copy of P and, relabeling vertices if necessary, we have the required embedding.…”

“…The situation is very different for odd values of m. For odd integers m ≥ 5, the smallest known integer v such that any partial m-cycle system of order u can be embedded in an m-cycle system of order v is v = (2u +1)m [8] (for m = 3 a best possible result is known, see [2]). However, Lindner and Rodger [8] only give embeddings of partial m-cycle systems of order u into m-cycle systems of order v for integers v ≥ (2u +1)m such that v ≡ m (mod 2m).…”

Embedding Partial Steiner Triple Systems with Few Triples

“…Most existing results on embedding partial Steiner triple systems concern embeddings of order at least 2u + 1. In particular, a series of results [23,18,1,3] guaranteed the existence of progressively smaller embeddings, culminating in a complete proof of Lindner's conjecture in [4].…”

Enclosings of λ-fold 4-cycle systems

“…Interestingly, the enclosing problem for 3-cycle systems is yet to be completely solved, with various efforts contributing to the current state of knowledge (see [20], for example). There are also related results on embeddings of partial cycle systems that may be of interest to the reader (see [1,4,14,13,17,18,24], for example).…”