# Increasing paths in countable graphs

## TL;DR

This paper explores conditions under which countable graphs, hypergraphs, and directed graphs contain infinite or arbitrarily long increasing paths, extending classical results to more complex structures and labelings.

## Contribution

It generalizes classical theorems about infinite degrees and increasing paths to hypergraphs, directed graphs, and various labelings, providing new equivalent conditions and constructions.

## Key findings

- Hypergraph infinite degree condition equivalent to infinite increasing loose path
- Existence of labelings with no one-way infinite increasing paths
- Characterization of paths of arbitrary finite length in graphs

## Abstract

In this paper we study variations of an old result by M\"{u}ller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by positive integers we can always find an infinite increasing path. We study corresponding questions for hypergraphs and directed graphs. For example we show that the condition that a hypergraph contains a subhypergraph with infinite degrees is equivalent to the condition that any vertex labeling contains an infinite increasing loose path. We also find an equivalent condition for a graph to have a property that any vertex labeling with positive integers contains a path of arbitrary finite length, and we study related problems for oriented graphs and labelings with $\mathbb{Z}$ (instead of $\mathbb{N}$). For example, we show that for every simple hypergraph, there is a labelling of its edges by $\mathbb{Z}$ that forbids one-way infinite increasing paths.