# Bipartite graphs and monochromatic squares

## TL;DR

This paper proves a set-theoretic result about bipartite graphs of certain sizes, showing they always contain large homogeneous squares, either complete or independent, under specific cardinality conditions.

## Contribution

It establishes a new consistency result in set theory regarding the existence of large homogeneous squares in bipartite graphs of successor cardinal size.

## Key findings

- Every bipartite graph of size κ^+×κ^+ contains a clique or independent set of size τ×τ for each τ in κ^+
- The result holds under the assumption of certain set-theoretic consistency conditions
- Advances understanding of large homogeneous structures in infinite bipartite graphs

## Abstract

We prove that consistently every bipartite graph of size $\kappa^+\times\kappa^+$ contains either a clique or an independent subset of size $\tau\times\tau$ for every $\tau\in\kappa^+$, where $\kappa$ is a successor cardinal.