On Tricovers of Pairs by Quintuples: $v\equiv 0(mod4)$

Abstract

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$C_3(v)\ge \lceil \frac{v\lceil \frac{3(v-1)}{4}\rceil }{5}\rceil =B_3(v)$ , where $\lceil x\rceil$ is the least integer not less than $x$. It is shown here that if $v\equiv 0(\mathrm{mod}4)$ and $v\ge 8$, then $C_3(v)=B_3(v)$.