Ray-Chaudhuri–Wilson theorem

Let $0\le s\le k\le n$, and let $\mathcal{F}$ be a uniform set family of $k$-element subsets of $[n]$. If distinct members have at most $s$ possible intersection sizes, then the Ray-Chaudhuri–Wilson theorem givesFor $s\le \min \{k,n-k\}$, let $L$ be the allowed intersection sizes and use, over the rational numbers, the intersection polynomialsTheir evaluation on the family's characteristic vectors of sets is diagonal with nonzero diagonal, so they are linearly independent. The low-degree evaluation rank on a uniform layer bounds the space of their restrictions by $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$. If $n-k<s\le k$, the bound follows directly from $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n}{k}\right)\le \left(\genfrac{}{}{0ex}{}{n}{s}\right)$.^[ref]

All $s$-element subsets attain equality when $k=s$. The range restriction matters: all five singleton subsets of $[5]$ have only one intersection size, hence also at most five, but using the invalid choice $s=5>k=1$ would falsely bound five members by $\left(\genfrac{}{}{0ex}{}{5}{5}\right)=1$.