Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set

Theorem
$44$ is the largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset:


 * $\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$


 * $\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$


 * $\set {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}$


 * $\set {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}$