Comparable Elements in Integer Lattices

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $N\geq1$, write $[N]=\{1,\ldots,N\}$. Two distinct triples $s,t\in[N]^3$ are comparable in the source sense if at least two coordinates of $s-t$ are strictly positive, or at least two are strictly negative. This is a symmetric sign-pattern relation, rather than the usual coordinatewise partial order. A set is pairwise comparable when every pair of its distinct elements has that property.

Statement recorded in the supplied source.

Do there exist constants $\delta>0$ and $N_0$ such that, for every integer $N\geq N_0$ and every pairwise comparable set $S\subset[N]^3$, \[ |S|\leq N^{2-\delta}? \] [S2]

Short English statement

Can one save a fixed power below the quadratic bound for sets of triples in which every pair has two coordinate differences of the same strict sign?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 142 (GREEN-054). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 88 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.