On a theorem of Erdös and Szekeres

In this note we generalize a theorem of Erdös and Szekeres, which states that every sequence of real numbers of length n² + 1 has a monotone subsequence of length n + 1, for points in certain metric spaces (R^k, d), where d is a Minkowski metric. Three theorems are proved concerning preassigned numbers of points which must lie on the same geodesic of the space, the last of which characterizes the class of Minkowski spaces under discussion.