Some new multi-cell Ramsey theoretic results

Abstract:

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.