
Relations in a set
Hello,
Given is a positive integer $\displaystyle k$. Prove that out of every set of integers which has more than $\displaystyle 3^k$ elements one can pick out a subset $\displaystyle S$ with $\displaystyle k+1$ elements and the following quality:
for any two subsets $\displaystyle A,B\subseteq S$ the sum of all elements in A is different from the sum of all elements in B.
I've been trying to play with basesystem representation for a while but it leads me nowhere. Your help will be appreciated.

I'm not sure about this, but I do have an idea about how you could approach this. Maybe you can prove that there is a subset you can make such that all numbers are "out of reach". For example, for k = 3 and the set is all numbers up until 27, the numbers 1, 4, 9 and 27 satisfy S.
Maybe if you order them and pick the $\displaystyle 3^0$th, $\displaystyle 3^1$th, $\displaystyle 3^2$th, $\displaystyle 3^3$th, etc. ?

Ok, but these numbers don't have to start with 1 and increment by 1, they can be any numbers, including negatives.

Yeah, I know. Hence the "maybe if you order them".
I have no idea how to actually prove this, but this was something that sprang to mind.