Around the Eggleston Theorem

Abstract This work is motivated by the two classical theorems on inscribing rectangles and squares into large subsets of the plane: Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem, we prove that for every Borel subset of the plane with uncountably many vertical sections that are positive (with respect to measure or category) contains a rectangle $$P\times B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>×</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> where P is perfect and B is Borel and positive. We also present a variant of Eggleston Theorem regarding the $$\sigma -$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> ideal $$\mathcal {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> generated by closed sets of measure zero. Furthermore, we show that every comeager (resp. conull) subset of the plane contains a rectangle $$[T]\times H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> <mml:mo>×</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> , where T is a splitting tree containing a Silver tree and H is comeager (resp. conull). Additionally, we establish a joint generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $$[T]\times H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> <mml:mo>×</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> modulo diagonal, where T is a uniformly perfect tree, H is comeager (resp. conull) and $$[T]\subseteq H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> <mml:mo>⊆</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> .