On Tricovers of Pairs by Quintuples: \(v ≡0 (mod 4)\)

A tricover of pairs by quintuples of a \(v\)-set \(V\) is a family of \(5\)-subsets of \(V\) (called blocks) with the property that every pair of distinct elements from \(V\) 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 covering 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 least integer not less than \(x\). It is shown here that if \(v \equiv 0 \pmod{4}\) and \(v \geq 8\), then \(C_3(v) = B_3(v)\).