On the \(\lambda\)-design conjecture (Q1913999)

A \(\lambda\)-design is a family of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks intersect in exactly \(\lambda\) points and not all blocks have the same cardinality. Ryser's and Woodall's conjecture states that any \(\lambda\)-design can be obtained from a symmetric design by complementing with respect to a fixed block. This conjecture is known to be true for all \(\lambda\leq 9\) and when \(\lambda\) is prime (see this paper for references). In this paper, the authors take a different view. They approach it by considering values of \(v\) as opposed to values of \(\lambda\). The main result of this paper is that the conjecture is true if \(v=p+1\), \(2p+1\), or \(3p+1\) for some prime \(p\).