An example of Tateno disproving conjectures of Bonato-Tardif, Thomasse, and Tyomkyn

TL;DR

This paper revisits Tateno's 2008 thesis to rigorously construct locally finite trees and partial orders with any finite number of equimorphy classes, disproving several conjectures in the field.

Contribution

It provides the first rigorous constructions of trees and partial orders with arbitrary finite equimorphy classes, disproving multiple longstanding conjectures.

Findings

Constructed locally finite trees with any finite number of equimorphy classes

Disproved the Bonato-Tardif conjecture on equimorphy classes of trees

Disproved conjectures by Thomasse and Tyomkyn

Abstract

In his 2008 thesis, Tateno claimed a counterexample to the Bonato-Tardif conjecture regarding the number of equimorphy classes of trees. In this paper we revisit Tateno's unpublished ideas to provide a rigorous exposition, constructing locally finite trees having an arbitrary finite number of equimorphy classes; an adaptation provides partial orders with a similar conclusion. At the same time these examples also disprove conjectures by Thomasse and Tyomkyn.