  We study several cardinal, and ordinal--valued functions that are relatives
of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq
L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an
ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min
{mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the
(phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+,
omega}, such that for every chi >= kappa, T has the (phi', chi)-order
property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in
L_{kappa^+,omega}}.
