Minimum embedding of Steiner triple systems into -designs II

Abstract: A -design of order embeds a given Steiner triple system if there is a subset of points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that , and that when equality is met, such a minimum embedding of an STS() exists, except when . Equality only holds when . One natural question is: What is the smallest order such that some STS can be embedded into a -design of order ? We solve the problem for 7 of the 10 congruence classes modulo 30. [Copyright &y& Elsevier]