  Given a $k$-uniform hyper-graph, the E$k$-Vertex-Cover problem is to find the
smallest subset of vertices that intersects every hyper-edge. We present a new
multilayered PCP construction that extends the Raz verifier. This enables us to
prove that E$k$-Vertex-Cover is NP-hard to approximate within factor
$(k-1-\epsilon)$ for any $k \geq 3$ and any $\epsilon>0$. The result is
essentially tight as this problem can be easily approximated within factor $k$.
Our construction makes use of the biased Long-Code and is analyzed using
combinatorial properties of $s$-wise $t$-intersecting families of subsets.
