Erdős-Ko-Rado theorem from shadows
= Erdős-Ko-Rado theorem from shadows
{c}
For an <intersecting family> of $r$-sets with $n\geq2r$, the rank-$r$ <lower shadow> of its complement family is disjoint from the original family. Iterating the <Kruskal-Katona theorem> shows that exceeding $\binom{n-1}{r-1}$ original members would force this shadow to have more than $\binom{n-1}r$ members, contradicting the total number $\binom nr$ of $r$-sets.