In \cite[Serra, Vena, Extremal families for the Kruskal-Katona theorem]{sv21}, the authors have shown a characterization of the extremal families for the Kruskal-Katona Theorem. We further develop some of the arguments given in \cite{sv21} and give additional properties of these extremal families. F\"uredi-Griggs/M\"ors theorem from 1986/85 \cite{furgri86,mors85} claims that, for some cardinalities, the initial segment of the colexicographical is the unique extremal family; we extend their result as follows: the number of (non-isomorphic) extremal families strictly grows with the gap between the last two coefficients of the $k$-binomial decomposition. We also show that every family is an induced subfamily of an extremal family, and that, somewhat going in the opposite direction, every extremal family is close to being the inital segment of the colex order; namely, if the family is extremal, then after performing $t$ lower shadows, with $t=O(\log(\log n))$, we obtain the initial segment of the colexicographical order. We also give a ``fast'' algorithm to determine whether, for a given $t$ and $m$, there exists an extremal family of size $m$ for which its $t$-th lower shadow is not yet the initial segment in the colexicographical order. As a byproduct of these arguments, we give yet another characterization of the families of $k$-sets satisfying equality in the Kruskal--Katona theorem. Such characterization is, at first glance, less appealing than the one in \cite{sv21}, since the additional information that it provides is indirect. However, the arguments used to prove such characterization provide additional insight on the structure of the extremal families themselves.

翻译：在文献\cite[Serra, Vena, Kruskal–Katona定理的极值族]{sv21}中，作者给出了Kruskal–Katona定理极值族的刻画。我们进一步发展了\cite{sv21}中的部分论证，并给出了这些极值族的额外性质。1986/85年的Füredi–Griggs/Mörs定理\cite{furgri86,mors85}指出，对于某些基数，逆向字典序的初始段是唯一的极值族；我们将其结果推广如下：（非同构）极值族的数量随$k$-二项式分解中最后两个系数之间的差距严格增长。我们还证明了每个族都是某个极值族的诱导子族，并且在某种程度上相反地，每个极值族都接近于逆向字典序的初始段；即，若该族是极值族，则经过$t$次下阴影操作（其中$t=O(\log(\log n))$）后，我们得到逆向字典序的初始段。此外，我们给出了一种“快速”算法，用于判断对于给定的$t$和$m$，是否存在一个大小为$m$的极值族，其第$t$次下阴影尚未成为逆向字典序的初始段。作为这些论证的副产品，我们给出了在Kruskal–Katona定理中达到等式的$k$-集族的另一种刻画。初看之下，该刻画不如\cite{sv21}中的刻画那样引人注目，因为它提供的附加信息是间接的。然而，用于证明该刻画的论证为极值族本身的结构提供了额外见解。