On Covering of Pairs by Quintuples

Abstract

Let $V$ be a finite set of order $v$. A $(v,k,\lambda )$ covering design of index $\lambda$ and block size $k$ is a collection of $k$-element subsets, called blocks, such that every $2$-subset of $V$ occurs in at least $\lambda$ blocks. The covering problem is to determine the minimum number of blocks, $\alpha (v,k,\lambda )$, in a covering design. It is well known that $\alpha (v,k,\lambda )\ge \lceil \frac{v}{k}\lceil \frac{v-1}{k-1}.\lambda \rceil \rceil =\varphi (v,k,\lambda )$, where $\lceil x\rceil$ is the smallest integer satisfying $x\le \lceil x\rceil$. In this paper, we determine the value $\alpha (v,5,\lambda )$, with few possible exceptions, for $\lambda =3$, $v\equiv 2(\mathrm{mod}4)$ and $\lambda =9,10,v\ge 5$, and $\lambda \ge 11$, $v\equiv 2(\mathrm{mod}4)$.