VN-TST 2023 Problem 6

Let $n \ge 3$ be an integer and $S$ be a set of $n$ elements. Determine the largest integer $k_n$ such that: for each selection of $k_n$ $3-$ subsets of $S$ , there exists a way to color elements of $S$ with two colors such that none of the chosen $3-$ subset is monochromatic.