An Improved Bound on the Cardinality of the Minimal Pairwise Balanced Designs on 18 Points with Maximum Block Size 4

Abstract

The cardinality of the minimal pairwise balanced designs on $v$ elements with largest block size $k$ is denoted by $g^{(k)}(v)$. It is known that $31\le g^{(4)}(18)\le 33$. In this paper, we show that $g^{(4)}(18)\ne 31$.