  We prove theorems of the following form: if $A\subseteq {\mathbb R}^2$ is a
big set, then there exists a big set $P\subseteq {\mathbb R}$ and a perfect set
$Q\subseteq {\mathbb R}$ such that $P\times Q\subseteq A$. We discuss cases
where big set means: set of positive Lebesgue measure, set of full Lebesgue
measure, Baire measurable set of second Baire category and comeagre set. In the
first case (set of positive measure) we obtain the theorem due to Eggleston. In
fact we give a simplified version of the proof given by J. Cichon. To prove
these theorems we use Shoenfield's theorem about absoluteness for
$\Sigma^1_2$-sentences.
