Triangle-intersecting families of graphs

Ellis, David; Filmus, Yuval; Friedgut, Ehud

doi:10.4171/jems/320

Cited by 62 publications

(73 citation statements)

References 13 publications

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“…as an important component in the proof of a 'stability' version of the Simonovits-Sós conjecture [12]. We trust that the symmetric-group versions will prove useful too.…”

Section: Conjecturementioning

confidence: 94%

A quasi-stability result for dictatorships in S n

2014

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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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“…as an important component in the proof of a 'stability' version of the Simonovits-Sós conjecture [12]. We trust that the symmetric-group versions will prove useful too.…”

Section: Conjecturementioning

confidence: 94%

A quasi-stability result for dictatorships in S n

2014

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“…If G consists of t pairwise disjoint k-sets then equality holds in (35). Also, if F is k-uniform, intersecting and τ (F) = k then C k (F) ⊃ F .…”

Section: On the Other Hand Provided C(x) ̸ = ∅ We Have τ (G(x)) = T−1mentioning

confidence: 99%

Invitation to intersection problems for finite sets

Frankl

Tokushige

2016

Journal of Combinatorial Theory, Series A

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Abstract. Extremal set theory is dealing with families, F of subsets of an nelement set. The usual problem is to determine or estimate the maximum possible size of F, supposing that F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any r subsets in F have at least t elements in common, all the sizes of pairwise intersections belong to a fixed set, L of natural numbers, there are no s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bounds.

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“…Their famous conjecture from 1976 states that the largest family of subgraphs of K n such that any two of them contain a triangle in their intersection, is unique (up to isomorphism) and is the family consisting of all the subgraphs containing a fixed triangle. This beautiful and long-standing conjecture was proved by Ellis, Filmus, and Friedgut [3]. The unique optimum for this problem is a kernel structure, consisting of all the graphs containing a "common kernel," the fixed triangle.…”

Section: Introduction Apart From Scattered Examples Cited Inmentioning

confidence: 95%

Families of Graph-different Hamilton Paths

Körner

Messuti

Simonyi³

2012

SIAM J. Discrete Math.

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Let D ⊆ N be an arbitrary subset of the natural numbers. For every n, let M (n, D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph Kn such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M (n, D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erdős, Simonovits, and Sós. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging.

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Triangle-intersecting families of graphs

Cited by 62 publications

References 13 publications

A quasi-stability result for dictatorships in S n

A quasi-stability result for dictatorships in S n

Invitation to intersection problems for finite sets

Families of Graph-different Hamilton Paths

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