The motif problem

TL;DR

This paper investigates the maximum number of specific geometric configurations, called motifs, within point sets in Euclidean space, establishing bounds and connections to hypergraph theory.

Contribution

It introduces a generalized motif problem in higher dimensions and relates the maximum motif count to a linear programming framework from hypergraph theory.

Findings

Maximum of r^2 motifs in a set of r points.

Generalization of motifs to sequences in R^p.

Connection to linear programming problems in hypergraph theory.

Abstract

Fix a choice and ordering of four pairwise non-adjacent vertices of a parallelepiped, and call a motif a sequence of four points in R^3 that coincide with these vertices for some, possibly degenerate, parallelepiped whose edges are parallel to the axes. We show that a set of r points can contain at most r^2 motifs. Generalizing the notion of motif to a sequence of L points in R^p, we show that the maximum number of motifs that can occur in a point set of a given size is related to a linear programming problem arising from hypergraph theory, and discuss some related questions.