Partial Answer

I have found a solution for

$n=54$

With the following sets

$\{3, 48, 41, 15, 35, 4, 17, 11, 20, 54, 39, 50, 18, 9, 24, 30, 43, 26, 37, 2, 36 \}$, $\{46, 33, 22, 34, 49, 8, 52, 21, 6, 13, 45, 25, 53, 40, 5, 38, 7 \}$, $\{12, 10, 16, 32, 1, 31, 19, 47, 29, 42, 14, 44, 28, 51, 23, 27 \}$

Although, this is also just a lower bound.

P.S. Please feel free to check my sets.

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.