Rectangle Divided into Incomparable Subrectangles

Theorem
Let $R$ be a rectangle.

Let $R$ be divided into $n$ smaller rectangles which are pairwise incomparable.

Then $n \ge 7$.

The smallest rectangle with integer sides that can be so divided into rectangles with integer sides is $13 \times 22$.


 * 7IncomparableRectangles.png