All lambda-designs with \(\lambda = 2 p\) are type-1 (Q1841522)

A \(\lambda\)-design is a family of blocks \(B_1,B_2,\ldots,B_v \subset \{1,2,\ldots,v\}\) such that \(|B_i \cup B_j|=\lambda\) for all \(i \neq j\) but not all \(B_i\) have the same size. The \(\lambda\)-design conjecture states that all \(\lambda\)-designs are obtained from symmetric designs by some complementary procedure (such \(\lambda\)-designs are called type-1). The author proves the \(\lambda\)-design conjecture for \(\lambda=2p\), where \(p\) is a prime number. A proof of the \(\lambda\)-design conjecture for \(\lambda \leq 34\) by using methods developed in this paper is also announced.