A Distributive Lattice Connected with Arithmetic Progressions of Length Three

Abstract

Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1,\dots ,n\}$ 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.

arXiv comment25 pages, 1 figure. To appear in the Ramanujan Journal