We must have two disjoint pairs of elements such that their sum is $9.$ Total such pairs possible is $5.\; \{0,9\}, \{1,8\}, \{2,7\}, \{3,6\}, \{4,5\}$
So, we can apply pigeon hole principle, in worst case, select one element from each pair and then if we select $2$ more elements then we are guaranteed to have a set which contains two disjoint subsets of size two, $\{x_{1}, x_{2}\}$ and $\{y_{1}, y_{2}\},$ such that $x_{1} + x_{2} = y_{1} + y_{2} = 9.$