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.