The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets

Ahlswede, Rudolf; Khachatrian, Levon H.

doi:10.1006/jcta.1996.0092

Cited by 106 publications

(106 citation statements)

References 8 publications

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“…This leads to the upper bound in (1). However, the m sets form an s-wise t-intersecting family, which implies that every set has at least t elements in common with a fixed set, say the first one.…”

Section: Motivation Definitions and Resultsmentioning

confidence: 94%

“…Problem 1 is also of independent interest in extremal set theory. Related works dealing with the determination of the largest family in MI s (n, k, t) include [1,2,6,9,12]. We make use of two important concepts in the study of families in MI s (n, k, t), namely kernels and generating sets.…”

Section: Motivation Definitions and Resultsmentioning

confidence: 99%

“…We make use of two important concepts in the study of families in MI s (n, k, t), namely kernels and generating sets. First, we define the notion of a generating set of a family of subsets, as introduced by Ahlswede and Khachatrian [1,2]. Let us fix some notation.…”

Section: Motivation Definitions and Resultsmentioning

confidence: 99%

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Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration

Moura

1999

Journal of Combinatorial Theory, Series A

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A family A of k-subsets of an n-set is said to be s-wise t-intersecting ifFor fixed s, n, k, and t, let I s (n, k, t) denote the set of all such families. A family A # I s (n, k, t) is said to be maximal if it is not properly contained in any other family in I s (n, k, t). We show that for fixed s, k, t, there is an integer n 0 =n 0 (k, s, t), for which the maximal families in I s (n 0 , k, t) completely determine the maximal families in I s (n, k, t), for all n n 0 . We give a construction for maximal families in I s (n+1, k+1, t+1) based on those in I s (n, k, t). Finally, for s=2, we classify the maximal families for k=t+1, n t+2, t 1, and for k=t+2, n t+6, t 1. The concepts of kernels and generating sets of a family of subsets play an important role in this work. 1999Academic Press, Inc.

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Section: Motivation Definitions and Resultsmentioning

confidence: 94%

Section: Motivation Definitions and Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration

Moura

1999

Journal of Combinatorial Theory, Series A

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“…Можно считать, что ε принадлежит интервалу (8), иначе мы возвращаемся к простым случаям, разобранным ранее. Таким образом, n −2 ≪ λ < 1.…”

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Экстремальные Задачи Для Раскрасок Равномерных Гиперграфов

Шабанов¹

2007

Izv. RAN. Ser. Mat.

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Экстремальные задачи для раскрасок равномерных гиперграфовИсследуется классическая задача экстремальной теории гиперграфов, впервые поставленная П. Эрдешем. Согласно определению Эрдеша гипер-граф обладает свойством B, если существует такая двухцветная раскрас-ка множества его вершин, в которой все ребра гиперграфа многоцветны. Задача состоит в нахождении величины m(n), равной минимуму из всех таких m, для которых существует n-равномерный (каждое ребро содер-жит ровно n вершин) гиперграф, имеющий в точности m ребер и не об-ладающий свойством B. Рассмотрены более общие постановки проблемы, в том числе в случае многоцветных раскрасок, и введен ряд параметриче-ских свойств для гиперграфов. Получены оценки экстремальных величин, определенных на различных классах гиперграфов и являющихся анало-гами величины m(n).Библиография: 14 наименований.

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“…Ahlswede and Khachatrian [1] determined m 0 (n, k, 2,t) completely, extending the earlier results by Hilton-Milner [18] and Frankl [11]. Brace and Daykin [4] determined w 0 (n, 1/2, r, 1) and Frankl determined w 0 (n, 1/2, r,t) for r ≥ 5 and 1 ≤ t ≤ 2 r − r − 1; in both cases G 1 (n, r,t) has the maximum p-weight.…”

Section: G(n Rt) = {G ⊂ 2 [N] : G Is R-wise T-intersecting}mentioning

confidence: 99%