Trees, ladders and graphs

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 $H_{\omega,\omega+2}$.