Some Intersection Theorems for Systems of Finite Sets

Hilton, A. J. W.; Milner, E. C.

doi:10.1093/qmath/18.1.369

Cited by 385 publications

(300 citation statements)

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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}supporting

confidence: 81%

The Random Walk Method for Intersecting Families

Tokushige

Bolyai Society Mathematical Studies

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

confidence: 81%

The Random Walk Method for Intersecting Families

Tokushige

Bolyai Society Mathematical Studies

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“…We need a few results from extremal set theory, some classical and some more recent. The first result that we will need, due to Hilton and Milner [15], bounds the cardinality of a nontrivial uniform intersecting family. Writing A x for the subfamily of a family A that consists of those sets containing x, we have the following.…”

Section: Preliminariesmentioning

confidence: 99%

“…Such questions about the 'stability' of the Erdős-Ko-Rado theorem have received a great deal of attention. Perhaps the earliest stability result about the Erdős-Ko-Rado theorem was proved by Hilton and Milner [15], who determined how large a uniform intersecting family can be if one insists that the family is nontrivial. Furthering this line of research, Friedgut [13], Dinur and Friedgut [11], and Keevash and Mubayi [18] have shown that every 'large' uniform intersecting family is essentially trivial.…”

Section: Introductionmentioning

confidence: 99%

Transference for the Erdős–ko–rado Theorem

2015

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For natural numbers n, r ∈ N with n r , the Kneser graph K (n, r ) is the graph on the family of r -element subsets of {1, . . . , n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K (n, r ) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as r/n is bounded away from 1/2, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős-Ko-Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

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“…Lemma 2 (Hilton-Milner [7]). If m ≥ 2n, then the maximum size of an stable set in K(m, n) with no centre is equal to 1 +…”

Section: Introductionmentioning

confidence: 99%

k -tuple chromatic number of the cartesian product of graphs

Bonomo

Koch

Torres

et al. 2015

Electronic Notes in Discrete Mathematics

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A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G, χ k (G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(G H) = max{χ(G), χ(H)}. In this paper, we show that there exist graphs G and H such thatMoreover, we also show that there exist graph families such that, for any k ≥ 1, the k-tuple chromatic number of their cartesian product is equal to the maximum k-tuple chromatic number of its factors.

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Some Intersection Theorems for Systems of Finite Sets

Cited by 385 publications

References 0 publications

The Random Walk Method for Intersecting Families

The Random Walk Method for Intersecting Families

Transference for the Erdős–ko–rado Theorem

k -tuple chromatic number of the cartesian product of graphs

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