The asymptotic minimum regularity of maximal (2,3)-antichains

A maximal $(2,3)$-antichain on $n$ points is a family of subsets in which every member has size $2$ or $3$, no member contains another, and which is maximal under these conditions. Its regularity is the common number of members containing each point. The authors construct such antichains with regularity $(13n/14-2)$ for every $n\equiv 0\pmod{14}$. The asymptotic minimum-regularity conjecture. The smallest possible regularity of a maximal $(2,3)$-antichain on $n$ points is $(13/14+o(1))n^2$. This conjecture asserts that the first construction in the theorem is asymptotically optimal; determining the exact minimum regularity remains open.