Choiceless Ramsey theory of linear orders
The authors study Ramsey theory of linear orders without the axiom of choice. They mainly study the linear orders $\langle ^{\alpha}2,<_{\mathrm{lex}} \rangle$ for an infinite ordinal $\alpha$ and the lexicographical order $<_{\mathrm{lex}}$. If $\kappa$ is an infinite cardinal with $\kappa^{<\kappa} = \kappa$, then in ZFC we have $\langle ^{\kappa}2,<_{\mathrm{lex}} \rangle \not\rightarrow \langle ^{\kappa}2,<_{\mathrm{lex}}\rangle^1_2$. \par Here, the authors obtain the consistency of several partition relations that are incompatible with the axion of choice. So the derive partition relations for $\langle ^{\omega}2,<_{\mathrm{lex}}\rangle$ from the Baire property for all subsets of $^{\omega}2$. They show that the above assumption implies that $\langle ^{\omega}2,<_{\mathrm{lex}} \rangle \rightarrow \langle ^{\omega}2,<_{\mathrm{lex}} \rangle^2_n$ for all natural numbers $n$. \par Starting with a model $V$ of ZFC and $\kappa$ a regular cardinal, they show that there is a symmetric extension of $V$ by a $<\kappa$-closed $\kappa^+$-c.c. forcing in which $\langle ^{\kappa}2,<_{\mathrm{lex}}\rangle \rightarrow \langle ^{\kappa}2,<_{\mathrm{lex}}\rangle^2_n$. \par They have negative partition relations for triples and quadruples. They completely classify the partition relations $\langle ^{\omega}2,<_{\mathrm{lex}} \rangle \rightarrow (\bigvee_{\nu<\lambda} K_\nu, \bigvee_{\nu<\mu}M_\nu)^m$ for linear orders $K_\nu$, $M_\nu$ and $m\leq 4$ and $\langle ^{\omega}2,<_{\mathrm{lex}} \rangle \rightarrow (K,M)^n$ for linear orders $K$, $M$ and $n$ a natural number, which are consistent.
scientific article; zbMATH DE number 6820905
scientific article