Stable sets of maximal size in Kneser-type graphs

We prove that Boolean functions on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn, are close to being unions of cosets of point-stabilizers. We use this to give a natural proof of a stability result on intersecting families of permutations, originally conjectured by Cameron and Ku [6], and first proved in [10]. We also use it to prove a 'quasistability' result for an edge-isoperimetric inequality in the transposition graph on Sn, namely that subsets of Sn with small edge-boundary in the transposition graph are close to being unions of cosets of point-stabilizers.

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“…Larose and Malvenuto [29] independently found a different proof of Theorem 9. More recently, Wang and Zhang [35] gave a shorter proof.…”

Section: Definition We Say That a Family A ⊂ S N Is Intersecting If mentioning

confidence: 99%

A quasi-stability result for dictatorships in S n

2014

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“…Let us start with an atypical and even somewhat trivial example, just to rephrase the already cited results of [1] and [10] in our present terms.…”

Section: Examplesmentioning

confidence: 99%

“…With this notation, x and y are adjacent if the product xy −1 is not a derangement. This is the complement of the graph of permutations studied by Cameron-Ku [1] and Larose-Malvenuto [10], that is the Cayley graph of permutations with generators the derangements. It is obvious that ρ(G, n) = ρ(G, [n]) = (n − 1)!…”

Section: Examplesmentioning

confidence: 99%

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Pairwise colliding permutations and the capacity of infinite graphs

Körner

Malvenuto²

2006

SIAM J. Discrete Math.

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Abstract. We call two permutations of the first n naturals colliding if they map at least one number to consecutive naturals. We give bounds for the exponential asymptotics of the largest cardinality of any set of pairwise colliding permutations of [n]. We relate this problem to the determination of the Shannon capacity of an infinite graph and initiate the study of analogous problems for infinite graphs with finite chromatic number.

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“…Intersecting families of permutations were initiated by Deza and Frankl in [10]. Some recent work done on this problem and its variants can be found in [5,7,8,11,12,19,26,33,34,35,36,42]. The investigation of the Erdős-Ko-Rado property for graphs started in [23], and gave rise to [4,6,21,22,24,44].…”

Section: Introductionmentioning

confidence: 99%

An Analogue of the Hilton-Milner Theorem for Weak Compositions

Ku¹,

Wong²

2015

Bulletin of the Korean Mathematical Society

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Abstract. Let N 0 be the set of non-negative integers, and let P (n, l) denote the set of all weak compositions of n with l parts, i.e., P (n, l) = {(x 1 , x 2 , . . . , x l ) ∈ N l 0 :A family A ⊆ P (n, l) is said to be trivially t-intersecting if there is a t-set T of [l] = {1, 2, . . . , l} and elements ys ∈ N 0 (s ∈ T ) such that A = {u ∈ P (n, l) : u(j) = y j for all j ∈ T }. We prove that given any positive integers l, t with l ≥ 2t + 3, there exists a constant n 0 (l, t) depending only on l and t, such that for all n ≥ n 0 (l, t), if A ⊆ P (n, l) is non-trivially t-intersecting, then Moreover, equality holds if and only if there is a t-set T of [l] such thatwhere As = {u ∈ P (n, l) : u(j) = 0 for all j ∈ T and u(s) = 0}and q i ∈ P (n, l) with q i (j) = 0 for all j ∈ [l] \ {i} and q i (i) = n.

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Stable sets of maximal size in Kneser-type graphs

Cited by 102 publications

References 10 publications

A quasi-stability result for dictatorships in S n

A quasi-stability result for dictatorships in S n

Pairwise colliding permutations and the capacity of infinite graphs

An Analogue of the Hilton-Milner Theorem for Weak Compositions

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