Helly-type theorems for the ordering of the vertices of a hypergraph

TL;DR

This paper establishes a Helly-type theorem for the ordering of vertices in hypergraphs with boundary markings, showing that local orderings imply a global ordering under certain conditions, with applications to betweenness theory.

Contribution

It introduces a Helly-type theorem for agreeing linear orders in hypergraphs, extending the understanding of vertex orderings and betweenness axioms in combinatorics.

Findings

If all small subhypergraphs have an agreeing linear order, then the entire hypergraph does too.

The constant 2r-2 in the theorem is optimal and cannot be reduced.

Certain rules of agreement do not admit a Helly-type property.

Abstract

Let $H$ be a complete $r$-uniform hypergraph such that two vertices are marked in each edge as its `boundary' vertices. A linear ordering of the vertex set of $H$ is called an {\em agreeing linear order}, provided all vertices of each edge of $H$ lie between its two boundary vertices. We prove the following Helly-type theorem: if there is an {agreeing linear order} on the vertex set of every subhypergraph of $H$ with at most $2r-2$ vertices, then there is an agreeing linear order on the vertex set of $H$. We also show that the constant $2r-2$ cannot be reduced in the theorem. The case $r=3$ of the theorem has particular interest in the axiomatic theory of betweenness. Similar results are obtained for further $r$-uniform hypergraphs ($r\geq 3$), where one or two vertices are marked in each edge, and the linear orders need to satisfy various rules of agreement. In one of the cases we prove that no such Helly-type statement holds.