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 $ω_1$ without uncountable $ω$-connected subgraphs. Second, we build triangle free graphs of chromatic number $ω_1$ without subgraphs isomorphic to $H_{ω,ω+2}$.

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