= Solution
Fix a <prime number> $p>r$. Distinct members of an intersecting $r$-uniform family have intersection size in
$$
L=\{1,2,\ldots,r-1\}\subset\mathbb F_p,
$$
whereas every member has size $r\notin L$ modulo $p$. The <Frankl-Wilson theorem> with $|L|=r-1$ therefore gives
$$
|\mathcal A|\leq\binom n{r-1}.
$$
For fixed $r$,
$$
\binom n{r-1}
=\binom{n-1}{r-1}\frac n{n-r+1}
=\binom{n-1}{r-1}\left(1+O(n^{-1})\right).
$$
This is the asserted asymptotic weakening of the <Erdős-Ko-Rado theorem>.
Solved by gpt-5.6-sol high.
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