On Erdős–Ko–Rado for Random Hypergraphs II

Hamm, Arran

doi:10.1017/s0963548318000433

Cited by 9 publications

(18 citation statements)

References 32 publications

(50 reference statements)

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“…the first member of ϕ −1 i (S i , F i ) according to some order-so the specification costs nothing. This strangely helpful device is from [6].) The key (trivial) point here is that (given…”

Section: Preview and Objectivementioning

confidence: 99%

The number of maximal independent sets in the Hamming cube

Park¹

2019

Preprint

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Let Qn be the n-dimensional Hamming cube and N = 2 n . We prove that the number of maximal independent sets in Qn is asymptotically 2n2 N/4 , as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and R ödl.The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them stability results for maximal independent set counts and old and new results on isoperimetric behavior in Qn.

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Section: Preview and Objectivementioning

confidence: 99%

The number of maximal independent sets in the Hamming cube

Park¹

2019

Preprint

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“…Beginning with the work of Balogh, Bohman and Mubayi [3], the problem of developing a 'random' version of the Erdős-Ko-Rado theorem has received significant attention (see [3,4,20,21,22]). In this section, we raise the analogous question for Katona's intersection theorem.…”

Section: A Random Version Of Katona's Intersection Theoremmentioning

confidence: 99%

Applications of graph containers in the Boolean lattice

Balogh

Treglown

Wagner

2016

Random Struct Algorithms

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We apply the graph container method to prove a number of counting results for the Boolean lattice P(n). In particular, we: Give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes in P(n), and we also give an upper bound on the number of transportation codes; Provide an alternative proof of Kleitman's theorem on the number of antichains in P(n) and give a two‐coloured analogue; Give an asymptotic formula for the number of (p, q)‐tilted Sperner families in P(n); Prove a random version of Katona's t‐intersection theorem. In each case, to apply the container method, we first prove corresponding supersaturation results. We also give a construction which disproves two conjectures of Ilinca and Kahn on maximal independent sets and antichains in the Boolean lattice. A number of open questions are also given. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 845–872, 2016

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“…While this work has been under review, there has been a vivid interest in questions related to random versions of the Erdős-Ko-Rado theorem (see [4,5,7,9,10,14,15]). In particular, as well as the results of Balogh, Bohman and Mubayi [3], the question concerning the structure of the largest intersecting family in the random setting has been addressed in [5,14,15] for various ranges of k and p. Moreover, an extension of the robust stability result for intersecting families, Lemma 2.3, has been considered in [9], implying that Theorem 1.3 can be extended to a larger range of k. We refer to these papers for further information.…”

Section: Discussionmentioning

confidence: 99%

Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

Gauy

Hàn

Oliveira

2017

Combinator. Probab. Comp.

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We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random kuniform hypergraph H k (n, p). For 2 ≤ k(n) ≤ n/2, let N = n k and D = n−k k . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of H k (n, p) has size (1 + o(1))p k n N , for any p ≫ n k ln 2 n k D −1 . This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well.A different behavior occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D −1 ≪ p ≤ (n/k) 1−ε D −1 , the largest intersecting subhypergraph of H k (n, p) has size Θ(ln(pD)N D −1 ), provided that k ≫ √ n ln n. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in H k (n, p), for essentially all values of p and k.

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On Erdős–Ko–Rado for Random Hypergraphs II

Cited by 9 publications

References 32 publications

The number of maximal independent sets in the Hamming cube

The number of maximal independent sets in the Hamming cube

Applications of graph containers in the Boolean lattice

Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

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