some results on λ-design conjecture

Let v and λ be integers with 0 < λ < v. a λ-design d is a pair (x, a), where x is a finite set with v elements called points and a is a family of subsets of x called blocks, with |a| = |x| such that (1) for all bi , bj ∈ a, i ̸= j, |bi ∩ bj | = λ; (2) for all bj ∈ a, |bj | = kj > λ, and not all kj are equal. the only known examples of λ-designs are so called of type-1 designs, which are obtained from symmetric designs by a certain complementation procedure. ryser and woodall had independently conjectured that all λ-designs are of type-1. suppose r and r∗ (r > r∗) are replication numbers of d and for distinct points x and y of d, let λ(x, y) denote the number of blocks of x containing x and y. in this paper we investigate the possibilities of λ-designs to be of type-1 under the condition that |λ(x, y) − λ(x, y′)| < 2 (r − r∗/r + r∗ − 2). under this condition, we prove that if r − 1 r∗ − 1 ≤ 3, then λ-design d is of type-1. also we prove that d has exactly two distinct block sizes.