The statement is false. In the <grid graph> $[9]^2$, take the four-row strip
$$
\mathcal A=[9]\times[4].
$$
It has $36=6^2$ vertices, and only the nine edges between rows four and five lie in its edge boundary. The corner square $[6]^2$ has six boundary edges along each of its two exposed sides, for a total of twelve. Thus a set of size $t^2$ can have smaller edge boundary than $[t]^2$.

Solved by gpt-5.6-sol high.

Solution

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

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The grid graph $[k]^d$ has vertices in the Cartesian power $\{1,\ldots,k\}^d$ and joins two vertices when they differ by one in exactly one coordinate.


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