Hajnal--M\'at\'e graphs, Cohen reals, and disjoint type guessing
Abstract
A Hajnal--M\'at\'e graph is an uncountably chromatic graph on ω1 satisfying a certain natural sparseness condition. We investigate Hajnal-M\'at\'e graphs and generalizations thereof, focusing on the existence of Hajnal-M\'at\'e graphs in models resulting from adding a single Cohen real. In particular, answering a question of D\'aniel Soukup, we show that such models necessarily contain triangle-free Hajnal-M\'at\'e 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 ZFC and, if disjoint type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal-M\'at\'e graphs G such that the chromatic numbers of finite subgraphs of G grow arbitrarily slowly.
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