How do I show with the pigeonhole principle that no seven positive integers not exceeding $24$ can have sums of all subsets different.
As observed by Ross Millikan, the simplest possible approach does not work. The maximum possible subset sum is $24+23+\cdots+18=147$, and the minimum possible subset sum is $0$. So as there are only $2^7=128$ subsets the pigeonhole principle does not force a collision. At least not without some sharpening.