Given a set $S=\{1,2,3,\ldots,3n\},(n\in N^*)$, let $T$ be a subset of $S$, such that for any $x, y, z\in T$ (not necessarily distinct) we have $x+y+z\not \in T$. Find the maximum number of elements $T$ can have.
Problem
Source: China south east mathematical olympiad 2008 day1 problem 1
Tags: combinatorics unsolved, combinatorics, Set systems, Additive combinatorics, Ramsey Theory