The set of values of b for which there exist block designs with b blocks

Let B denote the set of values of b for which there exists a block design with b blocks and for k⩾3, let Bk denote the subset of B determined by the designs with block size k. We present some information about B and the sets Bk. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k′ such that the equation b′=b+h has infinitely many solutions b,b′ satisfying b∈Bk and b′∈Bk′. The study is restricted to the case λ=1.