Trees, ladders and graphs

TL;DR

This paper presents a novel method for constructing uncountably chromatic graphs using non special trees and ladder systems, answering longstanding questions about graph connectivity and subgraph structures.

Contribution

It introduces a new construction technique for uncountably chromatic graphs and addresses open problems related to connectivity and subgraph configurations.

Findings

Constructed graphs of chromatic number ω₁ without uncountable ω-connected subgraphs.

Built triangle-free graphs of chromatic number ω₁ lacking specific subgraphs.

Provided answers to questions posed by Erdős and Hajnal in 1985.

Abstract

We introduce a new method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erd\H{o}s and A. Hajnal from 1985, we construct graphs of chromatic number $\omega_1$ without uncountable $\omega$-connected subgraphs. Second, we build triangle free graphs of chromatic number $\omega_1$ without subgraphs isomorphic to $H_{\omega,\omega+2}$.