In this post we analysed sums of the form

We will generalise this to sums of the form

for suitable functions .

**Theorem.** For , define . Suppose that satisfies

for some . Then there exist disjoint non-empty subsets such that

.

*First Proof.* Each non-empty subset corresponds to a sum

.

Since —the total number of non-empty subsets —there must be two non-empty subsets with . Excluding the common elements from both sides gives the desired claim.

*Second Proof.* We consider expressions of the form

,

where for each . Then takes possible values in , which are all congruent modulo . Hence, the number of distinct values that takes is at most

.

Since this is less than , we can find for such that

with for some .

Note that for all . Let and . Then and are non-empty and disjoint, , and

,

as desired.

*Remark.* This is tight: taking shows that

and

for any two disjoint non-empty subsets .