Weighted multiply intersecting families. (English) Zbl 1045.05084

Frankl, P.; Tokushige, N.

Stud. Sci. Math. Hung. 40, No. 3, 287-291 (2003).

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.

Reviewer: Péter L. Erdős (Budapest)

MSC:

05D05

Extremal set theory

Keywords:

intersecting family