Partial Answer: Lower Bound
I have found a solution for
$n=60$
With the following sets
$\{45, 49, 43, 58, 7, 20, 1, 11, 56, 22, 39, 30, 26, 50, 13, 33, 47, 37, 41, 46, 52, 60, 28 \}$, $\{55, 5, 29, 3, 57, 23, 34, 38, 32, 36, 19, 27, 53, 42, 25, 12, 14, 40, 10, 18, 21, 8, 16, 51 \}$, $\{9, 24, 31, 4, 15, 6, 35, 54, 59, 2, 17, 44, 48 \}$
Although, this is also just a lower bound.
P.S. Please feel free to check my sets.