  We study the entanglement properties of two-spin subsystems in spin-singlet
states. The average entanglement between two spins is maximized in a single
valence-bond (VB) state. On the other hand, $E_v^2$ (the average entanglement
between a subsystem of two spins and the rest of the system) can be maximized
through a homogenized superposition of the VB states. The maximal $E_v^2$
rapidly increases with system size and saturates at its maximum allowed value.
We adopt two ways of obtaining maximal $E_v^2$ states: (1) imposing homogeneity
on singlet states; and (2) generating isotropy in a general homogeneous state.
By using these two approaches, we construct explicitly four-spin and six-spin
highly entangled states that are both isotropic and homogeneous. Our maximal
$E^2_v$ states represent a new class of resonating-valence-bond states which we
show to be the ground states of the infinite-range Heisenberg model.
