  Recently, it was discovered by several authors that a $q$-ary optimal locally
recoverable code, i.e., a locally recoverable code archiving the Singleton-type
bound, can have length much bigger than $q+1$. This is quite different from the
classical $q$-ary MDS codes where it is conjectured that the code length is
upper bounded by $q+1$ (or $q+2$ for some special case). This discovery
inspired some recent studies on length of an optimal locally recoverable code.
It was shown in \cite{LXY} that a $q$-ary optimal locally recoverable code is
unbounded for $d=3,4$. Soon after, it was proved that a $q$-ary optimal locally
recoverable code with distance $d$ and locality $r$ can have length
$\Omega_{d,r}(q^{1 + 1/\lfloor(d-3)/2\rfloor})$. Recently, an explicit
construction of $q$-ary optimal locally recoverable codes for distance $d=5,6$
was given in \cite{J18} and \cite{BCGLP}.
  In this paper, we further investigate construction optimal locally
recoverable codes along the line of using parity-check matrices. Inspired by
classical Reed-Solomon codes and \cite{J18}, we equip parity-check matrices
with the Vandermond structure. It is turns out that a parity-check matrix with
the Vandermond structure produces an optimal locally recoverable code must obey
certain disjoint property for subsets of $\mathbb{F}_q$. To our surprise, this
disjoint condition is equivalent to a well-studied problem in extremal graph
theory. With the help of extremal graph theory, we succeed to improve all of
the best known results in \cite{GXY} for $d\geq 7$. In addition, for $d=6$, we
are able to remove the constraint required in \cite{J18} that $q$ is even.
