You are given a subset \(S\) of \(\{1,2,...,100\}\) with the following property:
for every quadruple \(u,v,w,x\) of distinct numbers in \(S\), the sum of \(u\) and \(v\) 
differs from the sum of \(w\) and \(x\).  Must the size of \(S\) be at most fifteen?
Prove your answer.