Cyclic interval intersection bound

An intersecting family of cyclic intervals of length $r$ in a cyclic order of $n$ positions has at most $r$ members when $1\le r\le n/2$. Rotate one selected interval to end at position $n$. Intervals ending at $r,\dots ,n-r$ miss it. Pair the remaining endpoints, other than $n$, as $(j,j+n-r)$ for $1\le j\le r-1$; each pair represents two disjoint intervals, hence contributes at most one member. This gives $1+(r-1)=r$. Counting such intervals over all permutations proves the Erdős-Ko-Rado theorem.