A Hilton–Milner-type theorem and an intersection conjecture for signed sets

Borg, Peter

doi:10.1016/j.disc.2013.04.030

Cited by 5 publications

(3 citation statements)

References 24 publications

(43 reference statements)

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“…A proof of Theorem 2 appeared in a previous version of this paper (which can still be found on the arXiv), but we were since informed that Theorem 2 is actually a special case of a theorem proved a few years ago by Borg [6] concerning intersecting families of "signed sets". His proof follows basically the same approach.…”

Section: Thenmentioning

confidence: 96%

Non-trivially intersecting multi-part families

Kwan

Sudakov

Vieira

2018

Journal of Combinatorial Theory, Series A

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We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially intersecting family in a natural "multi-part" setting.Here the ground set is divided into parts, and one considers families of sets whose intersection with each part is of a prescribed size. Our work is motivated by classical results in the singlepart setting due to Erdős, Ko and Rado, and Hilton and Milner, and by a theorem of Frankl concerning intersecting families in this multi-part setting. In the case where the part sizes are sufficiently large we determine the maximum size of a non-trivially intersecting multi-part family, disproving a conjecture of

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Section: Thenmentioning

confidence: 96%

Non-trivially intersecting multi-part families

Kwan

Sudakov

Vieira

2018

Journal of Combinatorial Theory, Series A

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“…In [6], Borg determined the structure of the largest non-trivial 1-intersecting subfamilies of L n,r,k . Our second main result extends Borg's result.…”

Section: Introductionmentioning

confidence: 99%

Large non-trivial $t$-intersecting families for signed sets

Yao¹,

Lv²,

Wang³

2021

Preprint

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For positive integers n, r, k with n r and k 2, a set {(x 1 , y 1 ), (x 2 , y 2 ), . . . , (x r , y r )} is called a k-signed r-set on [n] if x 1 , . . . , x r are distinct elements of [n] and y 1 . . . , y r ∈ [k]. We say a t-intersecting family consisting of k-signed r-sets on [n] is trivial if each member of this family contains a fixed k-signed t-set. In this paper, we determine the structure of large maximal non-trivial t-intersecting families. In particular, we characterize the non-trivial t-intersecting families with maximum size for t 2, extending a Hilton-Milner-type result for signed sets given by Borg.

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“…We should mention that there are several generalisations, extensions and variations of Theorem 1.1; see for example [2,3,5,6,8,11,21].…”

Section: Introductionmentioning

confidence: 99%

Intersecting families, signed sets, and injection

Feghali

2019

Preprint

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Let k, r, n ≥ 1 be integers, and let S n,k,r be the family of r-signed k-sets on [n] = {1, . . . , n} given byA well-known result (first stated by Meyer and proved using different methods by Deza and Frankl, and Bollobás and Leader) states that ifWe provide a proof of this result by injection (in the same spirit as Frankl and Füredi's and Hurlbert and Kamat's injective proofs of the Erdős-Ko-Rado Theorem, and Frankl's and Hurlbert and Kamat's injective proofs of the Hilton-Milner Theorem) whenever r ≥ 2 and 1 ≤ k ≤ n/2, leaving open only some cases when k ≤ n.

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A Hilton–Milner-type theorem and an intersection conjecture for signed sets

Cited by 5 publications

References 24 publications

Non-trivially intersecting multi-part families

Non-trivially intersecting multi-part families

Large non-trivial $t$-intersecting families for signed sets

Intersecting families, signed sets, and injection

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