Steiner t-designs for large t

TL;DR

This paper addresses the longstanding open problem in combinatorial design theory by demonstrating the non-existence of certain highly symmetric Steiner 6-designs, advancing understanding of the structure of Steiner t-designs.

Contribution

It proves that essentially no block-transitive Steiner 6-designs exist, extending previous results on flag-transitive designs and closing a gap in the classification of Steiner t-designs.

Findings

No non-trivial flag-transitive Steiner 6-design exists

No block-transitive Steiner 6-design exists

Advances understanding of symmetry constraints in Steiner t-designs

Abstract

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.