625 research outputs found

    An application of the Helly property to the partially ordered sets

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    AbstractQuilliot (Discrete Math. 1982.) showed that when the bowls of a connected graph satisfy the Helly property it is possible to deduce for this graph some fixed point and homomorphism extension theorems. For a partially ordered set E a special family of subsets is defined which, when it satisfies the Helly property, permits the deductions that every homomorphism from E into E has a fixed point, that every antitone function from E has “almost” a fixed point, and that there exists a simple criterion letting us know when a function f from a subset A of a partially ordered set G can be extended into a homomorphism from G to E

    Bounding Helly numbers via Betti numbers

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    We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers bb and dd there exists an integer h(b,d)h(b,d) such that the following holds. If F\mathcal F is a finite family of subsets of Rd\mathbb R^d such that β~i(G)b\tilde\beta_i\left(\bigcap\mathcal G\right) \le b for any GF\mathcal G \subsetneq \mathcal F and every 0id/210 \le i \le \lceil d/2 \rceil-1 then F\mathcal F has Helly number at most h(b,d)h(b,d). Here β~i\tilde\beta_i denotes the reduced Z2\mathbb Z_2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these d/2\lceil d/2 \rceil first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex KK, some well-behaved chain map C(K)C(Rd)C_*(K) \to C_*(\mathbb R^d).Comment: 29 pages, 8 figure

    Helly meets Garside and Artin

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    A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g.\ fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jones conjecture, the coarse Baum-Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a mean of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.Comment: Small modifications according to the referee report, updated references. Final accepted versio

    Core congestion is inherent in hyperbolic networks

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    We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network GG admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset XX of vertices of a δ\delta-hyperbolic graph GG there exists a vertex mm of GG such that the disk D(m,4δ)D(m,4 \delta) of radius 4δ4 \delta centered at mm intercepts at least one half of the total flow between all pairs of vertices of XX, where the flow between two vertices x,yXx,y\in X is carried by geodesic (or quasi-geodesic) (x,y)(x,y)-paths. A set SS intercepts the flow between two nodes xx and yy if SS intersect every shortest path between xx and yy. Differently from what was conjectured by Jonckheere et al., we show that mm is not (and cannot be) the center of mass of XX but is a node close to the median of XX in the so-called injective hull of XX. In case of non-uniform traffic between nodes of XX (in this case, the unit flow exists only between certain pairs of nodes of XX defined by a commodity graph RR), we prove a primal-dual result showing that for any ρ>5δ\rho>5\delta the size of a ρ\rho-multi-core (i.e., the number of disks of radius ρ\rho) intercepting all pairs of RR is upper bounded by the maximum number of pairwise (ρ3δ)(\rho-3\delta)-apart pairs of RR

    Quantitative Tverberg, Helly, & Carath\'eodory theorems

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    This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.Comment: 33 page

    Sunflowers of Convex Open Sets

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    A sunflower is a collection of sets {U1,,Un}\{U_1,\ldots, U_n\} such that the pairwise intersection UiUjU_i\cap U_j is the same for all choices of distinct ii and jj. We study sunflowers of convex open sets in Rd\mathbb R^d, and provide a Helly-type theorem describing a certain "rigidity" that they possess. In particular we show that if {U1,,Ud+1}\{U_1,\ldots, U_{d+1}\} is a sunflower in Rd\mathbb R^d, then any hyperplane that intersects all UiU_i must also intersect i=1d+1Ui\bigcap_{i=1}^{d+1} U_i. We use our results to describe a combinatorial code Cn\mathcal C_n for all n2n\ge 2 which is on the one hand minimally non-convex, and on the other hand has no local obstructions. Along the way we further develop the theory of morphisms of codes, and establish results on the covering relation in the poset PCode\mathbf P_{\mathbf{Code}}
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