A Distributive Lattice Connected with Arithmetic Progressions of Length Three

Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i, x_j, x_k$ and $x_a, x_b, x_c$ are arithmetic progressions. We determine the number of such collections $\mathcal{T}$ and the number of them of maximum size. These results confirm two conjectures of Noam Elkies.