  In this paper we investigate the sensitivity complexity of hypergraph
properties. We present a $k$-uniform hypergraph property with sensitivity
complexity $O(n^{\lceil k/3\rceil})$ for any $k\geq3$, where $n$ is the number
of vertices. Moreover, we can do better when $k\equiv1$ (mod 3) by presenting a
$k$-uniform hypergraph property with sensitivity $O(n^{\lceil k/3\rceil-1/2})$.
This result disproves a conjecture of Babai~\cite{Babai}, which conjectures
that the sensitivity complexity of $k$-uniform hypergraph properties is at
least $\Omega(n^{k/2})$. We also investigate the sensitivity complexity of
other symmetric functions and show that for many classes of transitive Boolean
functions the minimum achievable sensitivity complexity can be $O(N^{1/3})$,
where $N$ is the number of variables. Finally, we give a lower bound for
sensitivity of $k$-uniform hypergraph properties, which implies the {\em
sensitivity conjecture} of $k$-uniform hypergraph properties for any constant
$k$.
