Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Games and Ramsey-like cardinals

We generalise the $\alpha$-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals $\alpha$ to arbitrary ordinals $\alpha$, and answer several questions posed in that paper. In particular, we show that $\alpha$-Ramseys are downwards absolute to the core model $K$ for all $\alpha$ of uncountable cofinality, that strategic $\omega$-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic $\alpha$-Ramsey cardinals are equiconsistent with measurable cardinals for all $\alpha>\omega$. We also show that the $n$-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the $\alpha$-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).Comment: 33 pages, 2 figures. Added Theorem 4.20 saying that strategic $(\omega{+}1)$-Ramsey cardinals are equiconsistent with measurables, and fixed many typos. This version is forthcoming in the JS