# Maximal number of sets, obtained as intersections of semiintervals of $k$ linear orders

Given a finite set $$S$$ with $$n$$ elements, and a fixed small $$k$$ (say $$k=3$$), how to find $$k$$ linear orders $$\leq_1, \dots, \leq_k$$ on $$S$$, such that the number of feasible subsets of $$S$$ is asymptotically maximal. Here a set $$X\subseteq S$$ is feasible if it is obtained as an intersection of initial intervals of those orders, that is $$X = \bigcap_{i=1, \dots, k} \{x\leq_i x_i\}$$ for some $$\{x_i\}$$.

By asymptotically maximal I mean that the number of feasible subsets $$N_{n,k}=c_k n^k + o(n^k)$$ as $$n \rightarrow\infty$$, and that I am interested in $$c_k$$ (and in the structure of corresponding orders).