Generalising Fisher's inequality to coverings and packings

Abstract

In 1940 Fisher famously showed that if there exists a non-trivial $(v,k,λ)$-design then $λ(v-1) \geq k(k-1)$. Subsequently Bose gave an elegant alternative proof of Fisher's result. Here, we show that the idea behind Bose's proof can be generalised to obtain new bounds on the number of blocks in $(v,k,λ)$-coverings and -packings with $λ(v-1)<k(k-1)$.