Some acyclic systems of permutations are not realizable by triangulations of a product of simplices

TL;DR

This paper presents a counter-example to a conjecture in combinatorial geometry, showing that not all acyclic systems of permutations can be realized by triangulations of a product of simplices, challenging previous assumptions.

Contribution

It provides the first known counter-example to the acyclic system conjecture and explores necessary conditions related to the spread-out simplices conjecture.

Findings

Counter-example to the acyclic system conjecture.

Necessary conditions for potential counter-examples to the spread-out simplices conjecture.

Insights into the limitations of triangulations of products of simplices.

Abstract

The acyclic system conjecture of Ardila and Ceballos can be interpreted as saying the following: "Every triangulation of the 3-skeleton of a product of two simplices can be extended to a triangulation of the whole product". We show a counter-example to this. Motivation for this conjecture comes from a related conjecture, the "spread-out simplices" conjecture of Ardila and Billey. We give some necessary conditions that counter-examples to this second conjecture (if they exist) must satisfy.