Erdős–Ko–Rado Theorems: Algebraic Approaches

Godsil, Chris; Meagher, Karen

doi:10.1017/cbo9781316414958

Cited by 107 publications

(159 citation statements)

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“…Often Boolean degree 1 functions correspond to the largest families of intersecting objects, connecting them to Erdős-Ko-Rado (EKR) theorems (see [36]). Indeed, all our trivial examples are built from these intersecting families, which are the is indicator functions x + i .…”

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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“…We state a version of a result known as the clique-coclique bound. It is proved, for general association schemes, as Lemma 3.8.1 in [4]. Given an orthogonal basis for the Bose-Mesner algebra, we can compute orthogonal projections of matrices onto it-if M is an n k × n k matrix, its orthogonal projection Ψ(M) is given by Gram-Schmidt:…”

Section: Projections On To Matrix Algebrasmentioning

confidence: 99%

Using the existence of t-designs to prove Erdős–Ko–Rado

Godsil

Guo

2019

Discrete Mathematics

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“…Moreover, the partial geometry has the strict EKR star property if the stars are the only sets of intersecting lines of maximum size. It is known that any partial geometry has the EKR star property [13,Section 5.6].…”

Section: Preliminariesmentioning

confidence: 99%