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

Abstract

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