Partition theorems for systems of finite subsets of integers

We study generalizations of Ramsey theorem to systems of finite subsets of @w. A system L of finite subsets of @w is called to be Ramsey if for every partition L=L"[email protected]?L"2 there exists an infinite set [email protected][email protected] such that L"[email protected]?[Y]^<^@w=0 or L^[email protected]?[Y]^<^@w=0. We give some sufficient conditions for a system to be Ramsey. We also prove a theorem which concerns partitions into infinitely many classes. This may be regarded as a common generalization of Erdos-Rado and Nash-Williams theorems.