On Tricovers of Pairs by Quintuples: $v\equiv 1(mod4)$

Abstract

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 $C_3(v)\ge \lceil \frac{v\lceil \frac{3(v-1)}{4}\rceil }{5}\rceil =B_3(v)$, where $\lceil x\rceil$ is the smallest integer that is at least $x$. It is shown here that if $v\equiv 1(\mathrm{mod}4)$, then $C_3(v)=B_3(v)+1$ for $v\equiv 9$ or $17(\mathrm{mod}20)$, and $C_3(v)=B_3(v)$ otherwise.