Homotopy type of neighborhood complexes of Kneser graphs, $$\varvec{KG_{2,k}}$$ K G 2 , k

For integers $n\geq 1$, $k\geq 0$, the stable Kneser graph $SG_{n,k}$ (also called the Schrijver graph) has as vertex set the stable $n$-subsets of $[2n+k]$ and as edges disjoint pairs of $n$-subsets, where a stable $n$-subset is one that does not contain any $2$-subset of the form $\{i,i+1\}$ or $\{1,2n+k\}$. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number $k+2$. This article contains a study of the independence complexes of $SG_{n,k}$ for small values of $n$ and $k$. Our contributions are two-fold: first, we prove that the homotopy type of the independence complex of $SG_{2,k}$ is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to $SG_{n,2}$.

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The smooth case

10.23943/princeton/9780691161686.003.0012 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Homotopy Type ◽

Unit Vector ◽

Smooth Case ◽

Open Set ◽

Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

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Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

Discrete Mathematics ◽

10.1016/j.disc.2021.112430 ◽

2021 ◽

Vol 344 (7) ◽

pp. 112430

Author(s):

Johann Bellmann ◽

Bjarne Schülke

Keyword(s):

Short Proof ◽

Kneser Graphs

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On the neighborhood complex of s→-stable Kneser graphs

Discrete Mathematics ◽

10.1016/j.disc.2021.112302 ◽

2021 ◽

Vol 344 (4) ◽

pp. 112302

Author(s):

Hamid Reza Daneshpajouh ◽

József Osztényi

Keyword(s):

Neighborhood Complex ◽

Kneser Graphs

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Homotopy type-theoretic interpretations of constructive set theories

Mathematical Structures in Computer Science ◽

10.1017/s0960129519000148 ◽

2019 ◽

pp. 1-32

Author(s):

Cesare Gallozzi

Keyword(s):

Comparative Analysis ◽

Set Theory ◽

Homotopy Type ◽

Type Theory ◽

Axiom Of Choice ◽

Homotopy Type Theory ◽

Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

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