State and prove the local $L Y M$ inequality. Explain carefully when equality holds.

Define the colex order and state the Kruskal-Katona theorem. Deduce that, if $n$ and $r$ are fixed positive integers with $1 \leqslant r \leqslant n-1$, then for every $1 \leqslant m \leqslant\left(\begin{array}{l}n \\ r\end{array}\right)$ we have

$$\min \left\{|\partial \mathcal{A}|: \mathcal{A} \subset[n]^{(r)},|\mathcal{A}|=m\right\}=\min \left\{|\partial \mathcal{A}|: \mathcal{A} \subset[n+1]^{(r)},|\mathcal{A}|=m\right\}$$

By a suitable choice of $n, r$ and $m$, show that this result does not remain true if we replace the lower shadow $\partial A$ with the upper shadow $\partial^{+} \mathcal{A}$.