Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family \( {\user1{\mathcal{F}}} \subset 2^{{{\left[ n \right]}}} \) satisfies F1∩···∩F r ≠∅ for all F1, . . .,F r \( {\user1{\mathcal{F}}} \) and \( {\bigcap {_{{F \in {\user1{\mathcal{F}}}}} } }F = \emptyset \). We prove that there exists ε=ε(r) >0 such that \( {\sum {_{{F \in {\user1{\mathcal{F}}}}} } }\omega ^{{{\left| F \right|}}} {\left( {1 - \omega } \right)}^{{n - {\left| F \right|}}} \leqslant \omega ^{r} {\left( {r + 1 - r\omega } \right)} \) holds for 1/2≤w≤1/2+ε if r≥13.

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Correspondence to Peter Frankl.

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Frankl, P., Tokushige, N. Weighted Non-Trivial Multiply Intersecting Families. Combinatorica 26, 37–46 (2006). https://doi.org/10.1007/s00493-006-0003-4

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Mathematics Subject Classification (2000):

  • 05D05