$\lambda$-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 $v\le 90$ and $\lambda \ne 45$.