A Ramsey theorem for biased graphs

TL;DR

This paper proves a Ramsey-type theorem for biased graphs, showing that large complete biased graphs contain large complete subgraphs with highly symmetric balanced cycle structures, characterized via group-labellings.

Contribution

It introduces a Ramsey theorem for biased graphs, revealing structural patterns of balanced cycles in large complete biased graphs using group-labellings.

Findings

Large complete biased graphs contain large symmetric subgraphs.

Balanced cycles in these subgraphs have three specific structures.

These structures are described via group-labellings.

Abstract

A $biased\ graph$ is a pair $(G,\mathcal{B})$, where $G$ is a graph and $\mathcal{B}$ is a collection of `balanced' circuits of $G$ such that no $\Theta$-subgraph of $G$ contains precisely two balanced circuits. We prove a Ramsey-type theorem, showing that if $(G,\mathcal{B})$ is a biased graph which $G$ is a very large complete graph, then $G$ contains a large complete subgraph $H$ such that the set of balanced cycles within $H$ has one of three specific, highly symmetric structures, all of which can be described naturally via group-labellings.