A Hajnal--M\'{ a}t\'{ e} graph is an uncountably chromatic graph on $\omega_1$ satisfying a certain natural sparseness condition. We investigate Hajnal-M\'{ a}t\'{ e} graphs and generalizations thereof, focusing on the existence of Hajnal-M\'{ a}t\'{ e} graphs in models resulting from adding a single Cohen real. In particular, answering a question of D\'{ a}niel Soukup, we show that such models necessarily contain triangle-free Hajnal-M\'{ a}t\'{ e} 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 $\mathsf{ ZFC}$ and, if disjoint type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal-M\'{ a}t\'{ e} graphs $G$ such that the chromatic numbers of finite subgraphs of $G$ grow arbitrarily slowly.
Categories: math.LO math.CO