English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A Hajnal--M\'{a}t\'{e} graph is an uncountably chromatic graph on $\omega_1$satisfying a certain natural sparseness condition. We investigateHajnal-M\'{a}t\'{e} graphs and generalizations thereof, focusing on theexistence of Hajnal-M\'{a}t\'{e} graphs in models resulting from adding asingle Cohen real. In particular, answering a question of D\'{a}niel Soukup, weshow 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. Weshow that disjoint type guessing is independent of $\mathsf{ZFC}$ and, ifdisjoint type guessing holds in the ground model, then the forcing extension bya single Cohen real contains Hajnal-M\'{a}t\'{e} graphs $G$ such that thechromatic numbers of finite subgraphs of $G$ grow arbitrarily slowly.

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