An Erdős–Ko–Rado theorem for finite 2-transitive groups

Meagher, Karen; Spiga, Pablo; Tiep, Pham Huu

doi:10.1016/j.ejc.2016.02.005

Cited by 35 publications

(52 citation statements)

References 38 publications

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“…Indeed, all our trivial examples are built from these intersecting families, which are the is indicator functions x + i . In the group case, EKR theorems are known for all 2-transitive groups, and the largest examples are indeed Boolean degree 1 functions [46], so classifying all Boolean degree 1 functions on 2-transitive groups would be very interesting. One of the classical results in analysis of Boolean functions on the hypercube, the Friedgut-Kalai-Naor theorem [31], states that a Boolean function on the hypercube which almost has degree 1 (in the sense that it is close in L 2 norm to a degree 1 function, which is not necessarily Boolean) is close to a Boolean degree 1 function.…”

Section: Future Workmentioning

confidence: 99%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

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We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...

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Section: Future Workmentioning

confidence: 99%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

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“…The group PSU(3, q) has a twotransitive action on a set of size q 3 + 1, we will only consider this action. In [17] it is shown that PSU (3, q) has the EKR property, here we give more details of this result and further give a result about the largest intersecting sets in PSU (3, q). Namely, we prove that the characteristic vector for any maximum intersecting set in PSU(3, q) is a linear combination of characteristic vectors of the canonical cliques in PSU (3, q).…”

Section: Introductionmentioning

confidence: 85%

“…Equality holds since any canonical set meets this bound. This means that the group PSU(3, q) has the EKR property (this is shown in [17]). By Tables 3 and 4, provided that q = 5, only χ 2 gives the eigenvalue that gives equality in the equation for Hoffman's bound.…”

Section: Character Numbermentioning

confidence: 97%

“…There are many 2-transitive groups for which the eigenvalue corresponding to the character χ(g) = fix(g) − 1 is not the least eigenvalue of the derangement graph. In [17] it is shown that these groups do indeed have the EKR property; this is done using a weighted version of Hoffman's bound. For any graph a symmetric matrix A with rows and columns indexed by the vertices of the graph is said to be a weighted adjacency matrix of the graph if A u,v = 0 whenever u and v are not adjacent vertices.…”

Section: An Algebraic Proof Of Ekr Theoremsmentioning

confidence: 99%

“…The next step is to give a weighted adjacency matrix for Γ PSU(3,q) for which Hoffman's ratio bound does hold with equality. This weighting was given in [17].…”

Section: Character Numbermentioning

confidence: 99%

See 2 more Smart Citations

An Erdős-Ko-Rado theorem for the group $$\hbox {PSU}(3, q)$$ PSU ( 3 , q )

Meagher

2018

Des. Codes Cryptogr.

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In this paper we consider the derangement graph for the group PSU(3, q) where q is a prime power. We calculate all eigenvalues for this derangement graph and use these eigenvalues to prove that PSU(3, q) has the Erdős-Ko-Rado property and, provided that q = 2, 5, another property that we call the Erdős-Ko-Rado module property.2010 Mathematics Subject Classification. Primary 05C35; Secondary 05C69, 20B05.

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On the number of derangements and derangements of prime power order of the affine general linear groups

Spiga

2016

J Algebr Comb

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An Erdős–Ko–Rado theorem for finite 2-transitive groups

Cited by 35 publications

References 38 publications

Boolean degree 1 functions on some classical association schemes

Boolean degree 1 functions on some classical association schemes

An Erdős-Ko-Rado theorem for the group $$\hbox {PSU}(3, q)$$ PSU ( 3 , q )

On the number of derangements and derangements of prime power order of the affine general linear groups

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