\(\lambda\)-Designs with \(g =7\)

Abstract

A \(\lambda\)-design on \(v\) points is a set of \(v\) distinct subsets (blocks) of a \(v\)-set such that any two different 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 every \(\lambda\)-design can be obtained from a symmetric design by a certain complementation procedure. A result of Ryser and Woodall establishes that there exist two integers, \(r\) and \(r^*\), such that each point in a \(\lambda\)-design is in exactly \(r\) or \(r^*\) blocks. The main result of the present paper is that the \(\lambda\)-design conjecture is true for \(\lambda\)-designs with \(\gcd(r-1,r^*-1)=7\).