4 problems

Let $A_{(\alpha,n)}$ be the sets defined from $J_{\alpha}(\mathbb{R})$, and let a mouse set mean a set $A\subseteq\mathbb{R}$ of the form $A=\mathbb{R}\cap\mathcal{M}$ for some cou…

Rudominer's conjecture. If $[\beta,\beta]$ is a projective-like $\mathcal{S}$-gap which is not the successor of a strong $\mathcal{S}$-gap, then…

Assume $ZF + AD + DC + V=L(\mathbb{R})$. Let $\alpha\in ON$ and $n\in\omega$. For each $x\in\mathbb{R}$, define $\Gamma_x$ to be the collection of all pointsets $A$ such that for s…

Assume $ZF + AD + DC + V=L(\mathbb{R})$. A mouse set is a set $X\subseteq\mathbb{R}$ for which there is an $\omega_1+1$-iterable premouse $M$ with $X=M\cap\mathbb{R}$. Let $\Gamma$…