## Weighted multiply intersecting families.(English)Zbl 1045.05084

Let $${\mathcal F} \subset 2^{[n]}$$ be an $$r$$-wise intersecting set system. For a fixed parameter $$w$$ with $$0< w <1$$ let $$W_w ({\mathcal F})=\sum_{F\in {\mathcal F}} w^{| F| } (1-w)^{n-| F| }$$ be the weight of the family. The paper proves that in case of $$w \leq (r-1)/r$$ the weight of any $$r$$-wise intersecting family is at most $$w$$ and the equality holds for maximal trivially $$r$$-wise intersecting families. Furthermore in case of $$w > (r-1)/r$$ the equality $$\lim_{n\to \infty} \max_{\mathcal F} (W_w({\mathcal F})) = 1$$ holds.

### MSC:

 05D05 Extremal set theory

### Keywords:

intersecting family
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