Solution
= Solution
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.