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.