Around Eggleston Theorem

TL;DR

This paper extends classical theorems on inscribing rectangles into large sets in the plane, using advanced set-theoretic tools to establish new results about Borel and comeager sets containing structured rectangles.

Contribution

It proves new variants of Eggleston and Mycielski theorems using Shoenfield Absoluteness and introduces a generalization involving perfect trees and comeager sets.

Findings

Every uncountable Borel set with positive sections contains a perfect rectangle.

Every comeager or conull set contains a rectangle with a perfect tree and comeager set.

A unified theorem for large sets containing rectangles modulo diagonal.

Abstract

The motivation of this work are the two classical theorems on inscribing rectangles and squares into large subsets of the plane, namely Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem we prove that for every Borel subset of the plane with uncountably many positive (with respect to measure or category) vertical section contains a rectangle $P\times B$ where $P$ is perfect and $B$ is Borel and positive. We also obtained a variant of Eggleston Theorem regarding the $\sigma$-ideal $\mathcal(E)$ generated by closed sets of measure zero. Furthermore we proved that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$, where $T$ is a Spinas tree containing a Silver tree and $H$ is comeager (resp. conull). Moreover we obtained a common generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ modulo diagonal, where $T$ is a uniformly perfect tree, $H$ is comeager (resp. conull) and $[T]\subseteq H$.