On cross t-intersecting families of sets

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On cross t-intersecting families of sets

For all p, t with 0 < p < 0.11 and 1≤ t ≤ 1/(2p), there exists n0 such that for all n,k with n > n0 and k/n = p the following holds: if A and B are k-uniform families on n vertices, and |A∩B| ≥ t holds for all A ∈A and B ∈B, then |A ||B| ≤ (n−t k−t )2 .

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On Cross-intersecting Families of Sets

A family A of ‘-element subsets and a family B of k-element subsets of an n-element set are cross-intersecting if every set from A has a nonempty intersection with every set from B. We compare two previously established inequalities each related to the maximization of the product jAjjBj, and give a new and short proof for one of them. We also determine the maximum of jAjx‘ þ jBjxk for arbitrary...

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On r-Cross Intersecting Families of Sets

Let (r−1)n ≥ rk and let F1, . . . ,Fr ⊂ ([n] k ) . Suppose that F1∩ ·· ·∩Fr 6= / 0 holds for all Fi ∈ Fi, 1≤ i≤ r. Then we show that ∏i=1 |Fi| ≤ (n−1 k−1 )r .

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Cross-Intersecting Families of Labeled Sets

For two positive integers n and p, let Lp be the family of labeled n-sets given by Lp = { {(1, `1), (2, `2), . . . , (n, `n)} : `i ∈ [p], i = 1, 2 . . . , n } . Families A and B are said to be cross-intersecting if A ∩ B 6= ∅ for all A ∈ A and B ∈ B. In this paper, we will prove that for p > 4, if A and B are cross-intersecting subfamilies of Lp, then |A||B| 6 p2n−2, and equality holds if and o...

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Multiple cross-intersecting families of signed sets

A k-signed r-set on [n] = {1, ..., n} is an ordered pair (A, f), where A is an rsubset of [n] and f is a function from A to [k]. Families A1, ...,Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound t...

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ژورنال

عنوان ژورنال: Journal of Combinatorial Theory, Series A

سال: 2010

ISSN: 0097-3165

DOI: 10.1016/j.jcta.2010.01.002