Order property

A formula $\varphi (\bar{x},\bar{y})$ has the order property for $T$ when arbitrarily long tuples $(a_i),(b_j)$ exist with $\varphi (a_i,b_j)$ exactly when $i<j$. Compactness realizes this pattern along every linear order.

If a theory has the order property, a linear order with more than $\kappa$ cuts over a dense subset of size at most $\kappa$ produces more than $\kappa$ distinct types over that subset. Hence a theory with the order property is not $\kappa$-stable for any $\kappa \ge |T|$.