0586662 - MÚ 2025 RIV US eng J - Journal Article
Lambie-Hanson, Christopher - Uhrik, Dávid
Hajnal-Máté graphs, Cohen reals, and disjoint-type guessing.
Mathematika. Roč. 70, č. 3 (2024), č. článku e12261. ISSN 0025-5793. E-ISSN 2041-7942
R&D Projects: GA ČR(CZ) GA23-04683S
Institutional support: RVO:67985840
Keywords : Hajnal-Máté graphs * Cohen reals
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022
Method of publishing: Open access
https://doi.org/10.1112/mtk.12261

A Hajnal-Máté graph is an uncountably chromatic graph on (Formula presented.) 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 (Formula presented.) and, if disjoint-type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal-Máté graphs (Formula presented.) such that the chromatic numbers of finite subgraphs of (Formula presented.) grow arbitrarily slowly.
Permanent Link: https://hdl.handle.net/11104/0354095