  In this letter, we first construct a counter example to show that for any
given positive integer $K\geq 2$ and for any $\frac{1}{\sqrt{K+1}}\leq t<1$,
there always exist a $K-$sparse $\x$ and a matrix $\A$ with the restricted
isometry constant $\delta_{K+1}=t$ such that the OMP algorithm fails in $K$
iterations. Secondly, we show that even when
$\delta_{K+1}=\frac{1}{\sqrt{K}+1}$, the OMP algorithm can also perfectly
recover every $K-$sparse vector $\x$ from $\y=\A\x$ in $K$ iteration. This
improves the best existing results which were independently given by Mo et al.
and Wang et al.
