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 N \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 P f ( N ) \mathcal {P}_{f}(\mathbb {N}) be the set of finite nonempty subsets of N \mathbb {N} . Given any finite partition R {\mathcal R} of N \mathbb {N} , there exist B 1 B_1 , B 2 B_2 , A 1 , 2 A_{1,2} , and A 2 , 1 A_{2,1} in R {\mathcal R} and sequences ⟨ x 1 , n ⟩ n = 1 ∞ \langle x_{1,n}\rangle _{n=1}^\infty and ⟨ x 2 , n ⟩ n = 1 ∞ \langle x_{2,n}\rangle _{n=1}^\infty in N \mathbb {N} such that (1) for each F ∈ P f ( N ) F\in \mathcal {P}_{f}(\mathbb {N}) , ∑ t ∈ F x 1 , t ∈ B 1 \sum _{t\in F}x_{1,t}\in B_1 and ∑ t ∈ F x 2 , t ∈ B 2 \sum _{t\in F}x_{2,t}\in B_2 and (2) whenever F , G ∈ P f ( N ) F,G\in \mathcal {P}_{f}(\mathbb {N}) and max F > min G \max F > \min G , one has ∑ t ∈ F x 1 , t + ∑ t ∈ G x 2 , t ∈ A 1 , 2 \sum _{t\in F}x_{1,t}+\sum _{t\in G}x_{2,t}\in A_{1,2} and ∑ t ∈ F x 2 , t + ∑ t ∈ G x 1 , t ∈ A 2 , 1 \sum _{t\in F}x_{2,t}+\sum _{t\in G}x_{1,t}\in A_{2,1} . The partition R {\mathcal R} can be refined so that the cells B 1 B_1 , B 2 B_2 , A 1 , 2 A_{1,2} , and A 2 , 1 A_{2,1} must be pairwise disjoint.