Apply <coordinate compression in a product of paths> in both coordinates. Replacing every fiber by an initial interval preserves the number of edges inside that fiber; between two neighboring fibers, the compressed sets span the minimum of their sizes, at least their original intersection size. Repeating the operation therefore never decreases the number of edges spanned and eventually produces a down-set.

Write the nonzero row lengths of this down-set as
$$
a_1\geq a_2\geq\cdots\geq a_q>0,
\qquad \sum_{i=1}^q a_i=t^2.
$$
The horizontal edges number $t^2-q$, and the vertical edges number
$$
\sum_{i=2}^q a_i=t^2-a_1.
$$
Consequently
$$
e(\mathcal A)\leq2t^2-(q+a_1).
$$
Since $t^2\leq qa_1$, the <arithmetic-geometric mean inequality> gives $q+a_1\geq2t$. Hence
$$
e(\mathcal A)\leq2t^2-2t=2t(t-1)=e([t]^2),
$$
as required.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 109 / 3 / iii / Solution

Coordinate compression replaces each fiber parallel to one coordinate axis by an initial interval of the same size. It preserves edges inside fibers and cannot decrease edges between neighboring fibers, because compressed fibers have intersection equal to the smaller fiber size.


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