What is the largest possible relation on set?

Question

I know, that

The smallest possible reflexive relation on a non-empty set is the diagonal ordered pairs of Cartesian product.

The largest possible reflexive relation on a non-empty set is the entire Cartesian product.

Likewise, The smallest possible anti-symmetric relation on a non-empty set is a null set.

But, I am not able to discern...

The largest possible anti-symmetric relation on a non-empty set?

P.S. I am not asking about- Total number of possible anti-symmetric relation.

Answer 1

Let $A=\{1,2,3\}$. Then both $R=\{(1,1)\}$ and $S=\{(1,2), (1,3)\}$ are anti-symmetric but $R \not\subset S$ and $S \not\subset R$. So no largest anti-symmetric relation can exist.