Ubiquity and the Farey graph

TL;DR

This paper constructs a special countable planar graph demonstrating unique path properties, with implications for Ramsey theory, connectivity, and the Farey graph's characterization.

Contribution

It introduces a countable planar graph with controlled path multiplicity between vertices, advancing understanding of graph connectivity and Farey graph properties.

Findings

Constructed a countable planar graph with finite but arbitrarily many edge-disjoint paths between any two vertices.

Showed the graph's applications in Ramsey theory and connectivity studies.

Provided insights into the characterization of the Farey graph.

Abstract

We construct a countable planar graph which, for any two vertices $u,v$ and any integer $k\ge 1$, contains $k$ edge-disjoint order-compatible $u$--$v$ paths but not infinitely many. This graph has applications in Ramsey theory, in the study of connectivity and in the characterisation of the Farey graph.