A note on increasing paths in countable hypergraphs

TL;DR

This paper extends a known equivalence between infinite degree subgraphs and increasing paths from graphs to certain hypergraphs with finitely many Berge cycles, addressing a question about edge labelings.

Contribution

It confirms that the equivalence for edge labelings holds for linear hypergraphs with finitely many Berge cycles, generalizing previous vertex-labeling results.

Findings

Confirmed the equivalence for hypergraphs with finitely many Berge cycles.

Extended Reiterman's result from graphs to specific hypergraphs.

Addressed a question posed by Arman, Elliott, and R"odl.

Abstract

An old result of M\"uller and R\"odl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They asked whether an analogous equivalence holds for edge labelings, which Reiterman answered in the affirmative. Recently, Arman, Elliott, and R\"odl extended this problem to linear $k$-uniform hypergraphs $H$ and generalized the original equivalence for vertex labelings. They asked whether Reiterman's result for edge labelings can similarly be extended. We confirm this for the case where $H$ admits only finitely many Berge cycles.