On Covering of Pairs by Quintuples

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 \(\alpha(v, k, \lambda) \geq \left\lceil\frac{v}{k} \lceil \frac{v-1}{k-1}.\lambda \rceil \right\rceil=\phi(v, k, \lambda)\), where \(\lceil x \rceil\) is the smallest integer satisfying \(x\leq\lceil x \rceil\). In this paper, we determine the value \(\alpha(v,5,\lambda)\), with few possible exceptions, for \(\lambda = 3\), \(v \equiv 2 \pmod{4}\) and \(\lambda = 9, 10, v\geq5\), and \(\lambda \geq 11\), \(v \equiv 2 \pmod{4}\).