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

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 \( 31 \leq g^{(4)}(18) \leq 33 \). In this paper, we show that \( g^{(4)}(18) \neq 31 \).