Suppose we are given a set $S \subset \{1,2,\cdots,n\}$ with $|S|>n/2$. Is it true that $S$ contains some $a,b,c$ st. $a+b=c$?
A proof I saw does the following -
Consider the set $P =\{\max(S)-a:a\in S\}$. Clearly $|P| = |S| > n/2$. Therefore by pigeonhole there exists an $x\in P$ that is also in $S$. So $x=\max(S)-a'$ for some $a'$ in $S \implies \exists x,a'\in S : x+a'= \max(S)$.
This doesn't seem watertight, as the set $P$ contains a $0$, and therefore is not entirely contained in $\{1,2,\cdots,n\}$ so pigeonhole may not apply.
Is the original statement true? How do we prove it?
EDIT: What if $S\subset \{1,2,\cdots,n\}$ and $|S|>n/2+1$
The proof works for the $n \mod 2 = 0$ case. For $n \mod 2 = 1$, it works when $|S|> \lceil n/2\rceil$