  Let (A) and (B) be two first order structures of the same vocabulary. We
shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B
which we denote by G_{omega_1}(A,B). This game is like the ordinary
Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1
moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of
cardinality <= aleph_1. We prove the following results:
  Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such
that the game G_{omega_1}(A,B) is non-determined.
  Theorem B: If it is consistent that there is a measurable cardinal, then it
is consistent that G_{omega_1}(A,B) is determined for all A and B of
cardinality <= aleph_2.
  Theorem C: For any kappa >= aleph_3 there are A and B of cardinality kappa
such that the game G_{omega_1}(A,B) is non-determined.
