  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 $[T]\subseteq H$.
