Kruskal-Katona theorem
= Kruskal-Katona theorem
{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.