Characterization of finite colored spaces with certain conditions

TL;DR

This paper investigates properties of finite colored spaces, establishing an inequality between the number of equivalence classes of pairs and triples, and characterizing cases of equality.

Contribution

It proves that in finite colored spaces with at least five elements, the number of pairwise equivalence classes does not exceed that of triples, and explores the structure when equality holds.

Findings

Proves $a_2(r) \,\leq\, a_3(r)$ for spaces with at least five elements.

Characterizes the structure of spaces where $a_2(r) = a_3(r)$.

Provides conditions under which the equality case occurs.

Abstract

A colored space is the pair $(X,r)$ of a set $X$ and a function $r$ whose domain is $\binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Z\subseteq X$. We shall write $Y\simeq_r Z$ if there exists a bijection $f:Y\to Z$ such that $r(U)=r(f(U))$ for each $U\in\binom{Y}{2}$. We denote the numbers of equivalence classes with respect to $\simeq_r$ contained in $\binom{X}{2}$ and $\binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)\leq a_3(r)$ when $5\leq |X|$, and show what happens when the equality holds.