\(\lambda\)-Designs with Two Block Sizes

Abstract

A \(\lambda\)-design on \(v\) points is a set of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(4\)-designs can be obtained from symmetric designs by a complementation procedure. In this paper, we establish feasibility criteria for the existence of \(\lambda\)-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. We use these criteria and a computer to prove that the \(\lambda\)-design conjecture is true for all \(\lambda\)-designs with two block sizes with \(v \leq 90\) and \(\lambda \neq 45\).