Consistency of $\neg AC^{3}$ + $`χ(E_{G_{1}})=3$, $χ(E_{G_{2}})\geqω\impliesχ(E_{G_{1}\times G_{2}})=3$' and relative consistency via strongly compactness

Amitayu Banerjee, Zalán Gyenis

Published 2019-11-01Version 1

We prove Andr\'as Hajnal's \textbf{Theorem 2} of \cite{Haj1985} in different ways and observe a {\em permutation model} where the {\em axiom of choice for 3 element sets} fails but the statement in \textbf{Theorem 2} of \cite{Haj1985} still holds for $k=3$. We also observe that the {\em Dilworth's decomposition theorem for infinite p.o.sets of finite width} holds and a weaker form of {\L}o\'{s}'s lemma (p. 253 of \cite{HoRu1998}) fails in the permutation model of \textbf{Theorem 7} of \cite{HT2018} due to Lorenz Halbeisen and Eleftherios Tachtsis. Secondly, we weaken the large cardinal assumption of the results from \cite{AC2013} due to Arthur Apter and Brent Cody, from a supercompact cardinal to a strongly compact cardinal. Further, applying the {\em appropriate automorphism technique} from \cite{AH1991} we remove the additional assumption that {\em `every strongly compact cardinal is a limit of measurable cardinals'} from \textbf{corollary 2.32} of section 4, chapter 2 of \cite{Dim2011} by Ioanna Dimitriou.