On Tricovers of Pairs by Quintuples: \(v ≡ 1 (mod 4)\)

A tricover of pairs by quintuples on a \(v\)-element set \(V\) is a family of 5-element subsets of \(V\), called blocks, with the property that every pair of distinct elements of \(V\) occurs in at least three blocks. If no other such tricover has fewer blocks, the tricover is said to be minimum, and the number of blocks in a minimum tricover is the tricovering number \(C_3(v,5,2)\), or simply \(C_3(v)\). It is well known that \(C_3(v) \geq \lceil \frac{{v} \lceil \frac {3(v-1)}{4} \rceil} {5} \rceil = B_3(v)\), where \(\lceil x\rceil\) is the smallest integer that is at least \(x\). It is shown here that if \(v \equiv 1 \pmod{4}\), then \(C_3(v) = B_3(v) + 1\) for \(v \equiv 9\) or \(17 \pmod{20}\), and \(C_3(v) = B_3(v)\) otherwise.