  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.
