Martin Grüttmüller, Ian T. Roberts, Sue D'Arcy, Judith Egan
The minimum number of blocks in Pairwise Balanced Designs with maximum block size 4 - the final cases
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

05B30 Other designs, configurations, See also {51E30}

51E30 Other finite incidence structures, See also {05B30}

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 $g_{\lambda}^{(4)}(v)$ (or $g^{(4)}(v)$ when $\lambda = 1$).
All values of $g_{\lambda}^{(4)}(v)$ are known except for $\lambda = 1$ and $v = 17$ or $18$.
It is known that $30 \leq g^{(4)}(17)\leq 31$ and $32\leq g^{(4)}(18)\leq 33$.
In this paper we show that $g^{(4)}(17) \neq 30$ and $g^{(4)}(18) \neq 32$, thus finalising the determination of $g_{\lambda}^{(4)}(v)$ for all $\lambda$ and $v$.

Keywords: pairwise balanced design, perfect covering