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. Erdős and A. Hajnal from 1985, we construct graphs of chromatic number ω1ω1 without uncountable ω  -connected subgraphs. Second, we build triangle free graphs of chromatic number ω1ω1 without subgraphs isomorphic to Hω,ω+2Hω,ω+2.