Characterization of finite colored spaces with certain conditions

Mitsugu Hirasaka; Masashi Shinohara

paper

Characterization of finite colored spaces with certain conditions

arXiv:1802.06098

Abstract

A colored space is the pair $(X, r)$ of a set $X$ and a function $r$ whose domain is $(2 X)$ . Let $(X, r)$ be a finite colored space and $Y, Z \subseteq X$ . We shall write $Y ≃_{r} Z$ if there exists a bijection $f : Y \to Z$ such that $r (U) = r (f (U))$ for each $U \in (2 Y)$ . We denote the numbers of equivalence classes with respect to $≃_{r}$ contained in $(2 X)$ and $(3 X)$ 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.