Uniform set family ($\mathcal{A}\subseteq [n{]}^{(k)}$)

A $k$-uniform set family consists entirely of $k$-element subsets of a common ground set.

If $n\ge 2k$ and $\mathcal{A}\subseteq [n{]}^{(k)}$ is intersecting, then $|\mathcal{A}|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$.

Katona's circle method places a finite ground set in a uniformly counted cyclic order, proves a bound for the members of a set family that appear as cyclic intervals, and double-counts pairs of a member and a compatible cyclic order.