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arXiv · 2312.01828

Hajnal--M\'{a}t\'{e} graphs, Cohen reals, and disjoint type guessing

Chris Lambie-Hanson·Dávid Uhrik
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Abstract

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.

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Chris Lambie-Hanson, Dávid Uhrik. 2023-12-04. Hajnal--M\'{a}t\'{e} graphs, Cohen reals, and disjoint type guessing. https://arxiv.org/abs/2312.01828

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