1 Day to 2016 Final Call

Find the largest positive integer $k$ that satisfies the property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \ldots , A_{k}$ such that for all integers $n \ge 15$ and all $i \in \{\ 1,2, \ldots, k \}\$ , there exist two distinct elements of $A_{i}$ whose sum is $n$ .