On the covering number $g_1^{(4)}(18)$

Grannell, Mike; Griggs, Terry; Stanton, R. G. and Whitehead, C. A. (2005). On the covering number $g_1^{(4)}(18)$ . Utilitas Mathematica, 68, pp. 131–143.

Abstract

The minimum number of blocks having maximum size precisely four that are required to cover, exactly $\lambda$ times, all pairs of elements from a set of cardinality $v$ is denoted by $g_{\lambda}^{(4)}(v)$ . The values of $g_{\lambda}^{(4)}(v)$ are known apart from the cases $(v,\lambda)= (17,1)$ and $(18,1)$ . We prove that $g_1^{(4)}(18)\ge 32$ , thereby reducing this outstanding case to just two possible values, namely 32 and 33.