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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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