On the Determination of the Covering Numbers $C(2,5,v)$

Abstract

Let $V$ be a finite set of $v$ elements. A covering of the pairs of $V$ by $k$-subsets is a family $F$ of $k$-subsets of $V$, called blocks, such that every pair in $V$ occurs in at least one member of $F$. For fixed $v$, and $k$, the covering problem is to determine the number of blocks of any minimum (as opposed to minimal) covering. Denote the number of blocks in any such minimum covering by $C(2,k,v)$. Let $B(2,5,v)=\lceil v\lceil (v-1)/4\rceil /5\rceil$. In this paper, improved results for $C(2,5,v)$ are provided for the case $v\equiv 1$ $$ or $$ $2(mod4)$.$$ For $$ $v\equiv 2(mod4)$, $$ it $$ is $$ shown $$ that $C(2,5,270)=B(2,5,270)$ and $C(2,5,274)=B(2,5,274)$, establishing the fact that if $v\ge 6$ and $v\equiv 2mod4$, then $C(2,5,v)=B(2,5,v)$. In addition, it is shown that if $v\equiv 13(mod20)$, then $C(2,5,v)=B(2,5,v)$ for all but $15$ possible exceptions, and if $v\equiv 17(mod20)$, then $C(2,5,v)=B(2,5,v)$ for all but $17$ possible exceptions.