TITLE: Any $5$ element subset of $\{1,\dots, 9\}$ contains an arithmetic progression.
QUESTION [6 upvotes]: How to show that any $5$ element subset of $X={1,2,3,...9}$ contains at least one arithmetic progression of length $3$.
Just a hint would do. Please don't post complete solution.

REPLY [4 votes]: This statement can't be proven because it is false. Consider $$\{1,2,6,8,9\}$$