  We consider rather general spin-$1/2$ lattices with large coordination
numbers $Z$. Based on the monogamy of entanglement and other properties of the
concurrence $C$, we derive rigorous bounds for the entanglement between
neighboring spins, such as $C\leq 1/\sqrt{Z}$, which show that $C$ decreases
for large $Z$. In addition, the concurrence $C$ measures the deviation from
mean-field behavior and can only vanish if the mean-field ansatz yields an
exact ground state of the Hamiltonian. Motivated by these findings, we propose
an improved mean-field ansatz by adding entanglement.
