Consider the set . Prove that one can choose T to be a element subset of S such that none of the elements of T can be represented as the

Consider the set \(S = \{0, 1, 2, \ldots, 3^k-1\}\). Prove that one can choose T to be a \(2^k\) element subset of S such that none of the elements of T can be represented as the arithmetic mean of two distinct elements of T.