Dependence of ground-state energy of classical n-vector spins on n

We study the ground state energy E(G)(n) of N classical O(n) vector spins with the Hamiltonian H=-Sigma(i>j)J(ij)S(i).S(j) where the coupling constants {J(ij)} are arbitrary. We prove that E(G)(n) is independent of n for all n>n(max)(N)= left floor(sq rt[8N+1]-1)/2 right floor. We show that this bound is the best possible. We also derive an upper bound for E(G)(m) in terms of E(G)(n), for mj) J(ij) + E(G)(n)]/Sigma(i>j) J(ij). We describe a procedure for constructing a set of J(ij)'s such that an arbitrary given state, {S(i)}, is the ground state. We show that the problem of finding the ground state for the special case n=N is equivalent to finding the ground state of a corresponding soft-spin problem.