Some new multi-cell Ramsey theoretic results

We extend an old Ramsey Theoretic result which guarantees sums of terms from all partition regular linear systems in one cell of a partition of the set \mathbb{N} of positive integers. We were motivated by a quite recent result which guarantees a sequence in one set with all of its sums two or more at a time in the complement of that set. A simple instance of our new results is the following. Let \mathcal{P}_{f}(\mathbb{N}) be the set of finite nonempty subsets of \mathbb{N}. Given any finite partition {\mathcal{R}} of \mathbb{N}, there exist B_1, B_2, A_{1,2}, and A_{2,1} in {\mathcal{R}} and sequences \langle x_{1,n}\rangle _{n=1}^\infty and \langle x_{2,n}\rangle _{n=1}^\infty in \mathbb{N} such that (1) for each F\in \mathcal{P}_{f}(\mathbb{N}), \sum _{t\in F}x_{1,t}\in B_1 and \sum _{t\in F}x_{2,t}\in B_2 and (2) whenever F,G\in \mathcal{P}_{f}(\mathbb{N}) and \max F < \min G, one has \sum _{t\in F}x_{1,t}+\sum _{t\in G}x_{2,t}\in A_{1,2} and \sum _{t\in F}x_{2,t}+\sum _{t\in G}x_{1,t}\in A_{2,1}. The partition {\mathcal{R}} can be refined so that the cells B_1, B_2, A_{1,2}, and A_{2,1} must be pairwise disjoint.