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## Odd sets with even intersection

• Let $\mathcal{F}$ be a family of subsets of $[n]$ such that, for all $A, B \in \mathcal{F}$ with $A \neq B$, we have $|A \cap B|$ is even and $|A|$ is odd. Prove that $|\mathcal{F}| \leq n$.

• Let $\mathcal{F}$ be a family of subsets of $[n]$ such that, for all $A, B \in \mathcal{F}$ with $A \neq B$, we have $|A \cap B|$ is even and $|A|$ is even. Prove that $|\mathcal{F}| \leq 2^{\lfloor \frac{n}{2} \rfloor}$.

• Let $k$ be an integer, and let $\mathcal{F}$ be a family of subsets of $[n]$ such that $|A \cap B|=k$ for all $A,B \in \mathcal{F}$ with $A \neq B$. Prove that $|\mathcal{F}| \leq n$.

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