A Ramsey theorem for biased graphs

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.