  The problem of computing the chromatic number of Kneser hypergraphs has been
extensively studied over the last 40 years and the fractional version of the
chromatic number of Kneser hypergraphs is only solved for particular cases. The
\emph{$(p,q)$-extremal problem} consists in finding the maximum number of edges
on a $k$-uniform hypergraph $\mathcal{H}$ with $n$ vertices such that among any
$p$ edges some $q$ of them have no empty intersection. In this paper we have
found a link between the fractional chromatic number of Kneser hypergraphs and
the $(p,q)$-extremal problem and also solve the $(p,q)$-extremal problem for
graphs if $n$ is sufficiently large and $p \geq q \geq 3$ by proposing it as a
problem of extremal graph theory. With the aid of this result we calculate the
fractional chromatic number of Kneser hypergraphs when they are composed with
sets of cardinality 2.
