Saturation for Small Antichains

For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\text{sat}^*(n, \mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\leq 4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\text{sat}^*(n, \mathcal A_5)$ and $\text{sat}^*(n, \mathcal A_6)$. We also give some open problems inspired by our analysis.