Hajnal–Máté graphs, Cohen reals, and disjoint‐type guessing
Chris Lambie‐Hanson,
Dávid Uhrik
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 inte… Show more
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