1. A Local Version of Katona's Intersecting Shadow Theorem.
- Author
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Sales, Marcelo and Schülke, Bjarne
- Subjects
SHADOWING theorem (Mathematics) ,SET theory ,LOGICAL prediction ,FAMILIES - Abstract
Katona's intersection theorem states that every intersecting family F ⊆ [ n ] (k) satisfies | ∂ F | ⩾ | F | , where ∂ F = { F \ { x } : x ∈ F ∈ F } is the shadow of F . Frankl conjectured that for n > 2 k and every intersecting family F ⊆ [ n ] (k) , there is some i ∈ [ n ] such that | ∂ F (i) | ⩾ | F (i) | , where F (i) = { F \ { i } : i ∈ F ∈ F } is the link of F at i. Here, we prove this conjecture in a very strong form for n > k + 1 2 . In particular, our result implies that for any j ∈ [ k ] , there is a j-set { a 1 , ⋯ , a j } ∈ [ n ] (j) such that | ∂ F (a 1 , ⋯ , a j) | ⩾ | F (a 1 , ⋯ , a j) | . A similar statement is also obtained for cross-intersecting families. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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