  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.
