On Covering Pairs by Quintuples: The Cases $v\equiv 3$ or $11$ modulo $20$

Abstract

Let $C(v)$ denote the least number of quintuples of a $v$-set $V$ with the property that every pair of distinct elements of $V$ occurs in at least one quintuple. It is shown, for $v\equiv 3\text{ or }11\text{modulo}20$ and $v\ge 11$, that $C(v)=\lceil (v-1)/4\rceil$ with the possible exception of $v\in \{83,131\}$.