Baire class one colorings and a dichotomy for countable unions of $F_σ$ rectangles

Failures of the Silver Dichotomy in the Generalised Baire Space

We prove results that falsify Silver's dichotomy for Borel equivalence relations on the generalised Baire space under the assumption V=L.

Baire-class $ξ$ colorings: the first three levels

The $\mathbb{G}_0$-dichotomy due to Kechris, Solecki and Todor\vcević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb{G}_0$-dichotomy for $\boraxi$-measurable countable colorings when $ξ\leq 3$. A $\boraxi$-measurable countable coloring gives a covering of the diagonal consisting of countably many $\boraxi$ squares. This leads to the study of countable unions of $\boraxi$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $ξ\leq 2$.

广义贝尔空间的开放双超图二分法及其应用

The open dihypergraph dichotomy for generalized Baire spaces and its applications

研究广义贝尔空间的开放双超图二分法，通过两种方式将相关结果提升到${}^\kappa\kappa$，并应用此方法解决了关于广义贝尔空间可定义子集的多个问题，如胡列维茨二分法变体等。

The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_σ$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^ωω$. We consider the analogous statement (which we call Hurewicz dichotomy) for $Σ^1_1$ subsets of the generalized Baire space ${}^κκ$ for a given uncountable cardinal $κ$ with $κ=κ^{<κ}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) hol…

A dichotomy for Borel functions

The dichotomy discovered by Solecki in \cite{Sol} states that any Baire class 1 function is either $σ$-continuous or "includes" the Pawlikowski function $P$. The aim of this paper is to give an argument which is simpler than the original proof of Solecki and gives a stronger statement: a dichotomy for all Borel functions.

Laver Trees in the Generalized Baire Space

We prove that any suitable generalization of Laver forcing to the space $ κ^κ$, for uncountable regular $κ$, necessarily adds a Cohen $κ$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ κ^{<κ}=κ$, then every $<κ$-distributive tree forcing on $κ^κ$ adding a dominating $κ$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $κ$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08…