For my thesis I've been doing a lot of research concerning upwards closed sets and anti chains. A while ago while searching I thought I stumbled across a proof that gave an upper bound on the number of anti-chains/uppersets in a partially ordered set.

Now I'm not 100% sure someone has proven a fair upper bound (ofc n^2 is always true) and after hours of searching I can't seem to find it (anymore). I did come across some papers concerning upper bounds on special types antichains or upper bounds in special cases of partially ordered sets but I'm afraid they weren't near any level that I could comprehend.