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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-elementsubsets 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 ofthem of maximum size. These results confirm two conjectures of Noam Elkies.

A Distributive Lattice Connected with Arithmetic Progressions of Length Three