Dedekind's problem

Remarks

No closed formula of a practical kind is known, and the problem is open in this form. Exact values are known only for $n \le 9$; the value $M(9)$ was computed in 2023 with large computer searches. Simple bounds come from antichains: every family of $\floor{n/2}$-element subsets of $\set{1, \ldots, n}$ is an antichain, so $M(n) \ge 2^{\binom{n}{\floor{n/2}}}$. Kleitman proved that $\log_2 M(n) = (1 + o(1)) \binom{n}{\floor{n/2}}$ as $n \to \infty$, and finer asymptotic formulas are known.