Hajnal--Máté graphs, Cohen reals, and disjoint type guessing

A Hajnal–Máté graph is an uncountably chromatic graph on ${\omega }_1$ satisfying a certain natural sparseness condition. We investigate Hajnal-Máté graphs and generalizations thereof, focusing on the existence of Hajnal-Máté graphs in models resulting from adding a single Cohen real. In particular, answering a question of Dániel Soukup, we show that such models necessarily contain triangle-free Hajnal-Máté graphs. In the process, we isolate a weakening of club guessing called \emph{disjoint type guessing} that we feel is of interest in its own right. We show that disjoint type guessing is independent of $\mathsf{Z}\mathsf{F}\mathsf{C}$ and, if disjoint type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal-Máté graphs $G$ such that the chromatic numbers of finite subgraphs of $G$ grow arbitrarily slowly.