On 6-sparse Steiner triple systems

Abstract

We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907.

Further 6-sparse Steiner Triple Systems

“…The special mitres and Pasch configurations that are shown in [4] to be unavoidable in all systems with sufficiently large order obtained from Theorem 2 do not form in the twogenerator systems of Theorem 4. We now prove that there is no such blocking mechanism to prevent the formation of 6-sparse two-generator systems of arbitrarily large orders.…”

“…In [4] we presented the first known non-trivial examples of 6-sparse Steiner triple systems. Our results depended on two basic theorems.…”

“…Furthermore, by employing Weil's theorem on bounding character sums [11, page 43] we were able to show that our list of such systems is complete. The other theorem from [4] asserts that the standard product construction preserves 6-sparseness under certain conditions. Theorem 3.…”

“…Such configurations are of particular interest because of the following result proved in [4]. Theorem 1.…”

Nonexistence of sparse triple systems over abelian groups and involutions

“…The direct product construction for 5-sparse STSs, developed by Ling [27], employs an abelian group which acts regularly on the point set. Forbes, Grannell and Griggs [13] discovered a construction method for blocktransitive STSs and found examples of 6-sparse STSs, which have the highest sparseness at the time of writing. They also developed a recursive construction similar to Theorem 1.5 for 6-sparse STSs and constructed infinitely many examples of such STSs.…”