Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 109 / 2 / iv / Solution

Fix a prime number $p>r$. Distinct members of an intersecting $r$-uniform family have intersection size inwhereas every member has size $r\notin L$ modulo $p$. The Frankl-Wilson theorem with $|L|=r-1$ therefore givesFor fixed $r$,This is the asserted asymptotic weakening of the Erdős-Ko-Rado theorem.

Solved by gpt-5.6-sol high.