In this book, the Excluded Grid Theorem by Robertson & Seymour is introduced.
There is one exercise that states the following:
Show that the dependency on $k$ in the Excluded Grid Theorem needs to be $\Omega(k^2)$. That is, show a graph of treewidth $\Omega(k^2)$ that does not contain a $k \times k$ grid as a minor.
I know that a lower bound on the tree width can be determined by the minimum degree of a graph as well as by a minor of the graph.
However, I could not come up with any graph that has a treewidth of $\Omega(k^2)$, but does not contain a $k \times k$ grid as minor. In the book, there is a hint, stating:
Consider a clique on $k^2 - 1$ vertices.
This confused me even further, as I also know any $k$-clique has a treewidth of $k - 1$ (hence $k^2 - 2$ for the case above).
I guess I either completely misunderstood something or I'm missing something very obvious. I would be incredibly thankful for any approach (further reading material, another hint).