Mixing Properties and the Chromatic Number of Ramanujan Complexes

Evra, Shai; Golubev, Konstantin; Lubotzky, Alexander

doi:10.1093/imrn/rnv022

Cited by 22 publications

(35 citation statements)

References 16 publications

(11 reference statements)

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“…Lemma 4.6. [EGL,Corollary 3.4] Let G = (V 1 V 2 , E) be a bipartite biregular graph, and let λ(G) be its normalized second largest eigenvalue. Then, for any A ⊂ V 1 and any B ⊂ V 2 ,…”

Section: Skeleton Mixing Lemmamentioning

confidence: 99%

Bounded degree cosystolic expanders of every dimension

Evra

Kaufman

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.A family of d-dimensional simplicial complexes is said to have the topological overlapping property, if there exists c > 0, such that each member of the family has the c-topological overlapping property.Gromov then proved the remarkable result that, for a fixed d ∈ N, the family of complete ddimensional complexes have the topological overlapping property. This result is a very striking generalization of classical results from convex combinatorics due to Boros-Furedi [BF] (for d = 2) * Hebrew University, ISRAEL.

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“…Lemma 4.6. [EGL,Corollary 3.4] Let G = (V 1 V 2 , E) be a bipartite biregular graph, and let λ(G) be its normalized second largest eigenvalue. Then, for any A ⊂ V 1 and any B ⊂ V 2 ,…”

Section: Skeleton Mixing Lemmamentioning

confidence: 99%

Bounded degree cosystolic expanders of every dimension

Evra

Kaufman

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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“…For example they give the best (from a quantitative point of view) known examples of "high girth, high chromatic number" graphs. After finding the appropriate high dimensional notions of "girth" and "chromatic number", these results can indeed be generalized to the Ramanujan complexes constructed in [LSV05b], (see [LM07], [GP14], [EGL15]).…”

Section: High Dimensional Expanders: Spectral Gapmentioning

confidence: 99%

High Dimensional Expanders

Lubotzky

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf.[Lub94], [HLW06], [Lub12] and the references therein). In the last decade, a theory of "high dimensional expanders" has begun to emerge. The goal of the current paper is to describe some paths of this new area of study.

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“…• Mixing for partite Ramanujan complexes was proven by Evra, Golubev and Lubotzky [EGL15]. The technique of [EGL15] is very different from ours and relied on a quantitative version of property (T).…”

Section: Introductionmentioning

confidence: 98%

“…The technique of [EGL15] is very different from ours and relied on a quantitative version of property (T). The error term in [EGL15] is independent of the size of the sets, i.e., in [EGL15] the bound on |X(U 0 ,...,Un)| |X(n)| − |U 0 |...|Un| |S 0 |...|Sn| is indpenedent on |U 0 |, ..., |U n | and depends only on the thickness (or equivalently, the local spectral gap) of the complex. Our results improve on the work of [EGL15] regarding both generality (since it is purely combinatorial) and the error term.…”

Section: Introductionmentioning

confidence: 99%