I am reading the following paper : I. Balla, B. Bollobas, T. Eccles, Union-closed families of sets, Journal of Combinatorial Theory Series A, 120 (2013) 531-544.
In the second Theorem, the authors mention Kruskal-Katona Theorem and say that: "The fundamental theorem of Kruskal and Katona states that if $\mathcal A\subseteq\mathbb{N}^{(k)}$ with $|\mathcal A|=m$. then the lower shadow of $\mathcal A$ is at least as large as the lower shadow of $\mathcal I_{k}(m)$; consequently, $|\partial^i(\mathcal A)|\geq |\partial^i(\mathcal I_{m})|$ for all $\mathcal A\subseteq\mathbb{N}^{(k)}$ with $|\mathcal A|=m$ and $0\leq i\leq k$."
Here, $\mathcal I_k(m)$ is the set of intial segment of length $m$ of $\mathbb{N}^{(k)}$ in colex order. I have a couple of questions here:
What is the meaning of $\partial^i(\mathcal A)$? From what I could find online, lower shadow of a set $A$ is $\partial(A)=\{A-\{i\}:i\in [n]\}$ and lower shadow of a family is union of lower shadows of all the members. But what is $\partial^i(\mathcal A)$?
On the wikipedia page of the Kruskal Katona theorem, the statement of the theorem is given as: " An integral vector $(f_0, f_1, ..., f_{d-1})$ is the $f$-vector of some $(d-1)$-dimensional simplicial complex if and only if
: $ 0 \leq f_{i} \leq f_{i-1}^{(i)},\quad 1\leq i\leq d-1.$"
This, at a first glance and a few glances after that, looks far away from the statement provided in the paper. Are these two equivalent?