On a result of Frankl and Wilson

Question

In the paper 'Intersection theorems with geometric consequences' (Combinatorica 1981) P. Frankl and R. M. Wilson consider families $\mathcal{F}$ of $k$-subsets of $\{1,\dots,n\}$ with the restriction on mutual intersections: $|A\cap B|-k$ is not divisible by $q$, where $q$ is a fixed prime power, whenever $A\ne B$ are from $\mathcal{F}$. They claim that $|\mathcal{F}|\leqslant \binom{n}{q-1}$, it is cited also by R. L. Graham here. This bound looks very strange: what if $n=q-1>k$, for example? Or $q-1>n$?

I think that from their proof the following weaker estimate may be obtained: $|\mathcal{F}| \leqslant \sum_{i\leqslant q-1}\binom{n}{i}$.

Now the questions: was there errata, or maybe everything is correct and I miss something? What are current best lower and upper bounds in this question?

Answer 1

The form of the Frankl-Wilson that I know is the following.

Frankl-Wilson Theorem. Let $q$ be a prime power and let $\mathcal{F}$ be a family of $k$-sets of an $n$-set such that $|A \cap B| \equiv \lambda_1, \dots,$ or $\lambda_s$ for all distinct $A,B \in \mathcal{F}$ and where $\lambda_i \not\equiv k$ for all $i$. Then $|\mathcal{F}| \leq \binom{n}{s}$.

I believe this form of the theorem also appears in the original paper of Frankl and Wilson. It seems to me that there is an implicit assumption that each $\lambda_i$ actually does occur as the intersection number of two sets in $\mathcal{F}$. Otherwise, as you have said, it does not look like the theorem is technically correct. However, with the additional assumption, I think everything is fine. For example, with the additional assumption, we have that $s$ is clearly at most $k$, since each set in $\mathcal{F}$ is a $k$-set. Therefore, in the situation that $s=q-1$ (by taking the set of $\lambda_i$ to be all elements of $\mathbb{Z} / q\mathbb{Z}$ except $k$), we must also have $k \geq q-1$. So the situation you describe $n=q-1>k$ is impossible for $s=q-1$.

If we are considering the non-uniform version of the Frankl-Wilson Theorem (where the sets are allowed to have different sizes), then the bound that you prove is indeed the correct one. For example, see Theorem 1.3 of this paper of Alon, Babai and Suzuki.