On the covering number g 1 (4)(18)

Abstract

The minimum number of blocks having maximum size precisely four that are required to cover, exactly λ times, all pairs of elements from a set of cardinality v is denoted by g _λ ⁽⁴⁾(v). The values of g _λ ⁽⁴⁾(v) are known apart from the cases (v, λ) = (17, 1) and (18, 1). We prove that g ₁ ⁽⁴⁾(18) ≥ 32, thereby reducing this outstanding case to just two possible values, namely 32 and 33.