{c}
{wiki}

If
$$
|\mathcal A|=\binom{a_r}{r}+\binom{a_{r-1}}{r-1}+\cdots+\binom{a_s}{s},
\qquad a_r>\cdots>a_s\geq s,
$$
then
$$
|\partial\mathcal A|\geq
\binom{a_r}{r-1}+\binom{a_{r-1}}{r-2}+\cdots+\binom{a_s}{s-1}.
$$
Initial segments of <colexicographic order> attain equality.


Kruskal-Katona theorem

{wiki=Kruskal–Katona_theorem}

The Kruskal-Katona theorem is a result in combinatorial set theory, particularly related to the theory of hypergraphs and the study of families of sets. It provides a connection between the structure of a family of sets and the number of its intersections. The theorem defines conditions under which an antipodal family (a family of subsets) can be characterized in terms of its lower shadow, which is a fundamental concept in combinatorics.


 Kruskal–Katona theorem

ID: kruskal-katona-theorem