$\lambda$-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 $r^{\ast }$, such that each point in a $\lambda$-design is in exactly $r$ or $r^{\ast }$ blocks. The main result of the present paper is that the $\lambda$-design conjecture is true for $\lambda$-designs with $gcd(r-1,r^{\ast }-1)=7$.