An application of the Helly property to the partially ordered sets

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

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 $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ 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 $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

Helly meets Garside and Artin

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

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ 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 $X$ of vertices of a $\delta$-hyperbolic graph $G$ there exists a vertex $m$ of $G$ such that the disk $D(m,4 \delta)$ of radius $4 \delta$ centered at $m$ intercepts at least one half of the total flow between all pairs of vertices of $X$, where the flow between two vertices $x,y\in X$ is carried by geodesic (or quasi-geodesic) $(x,y)$-paths. A set $S$ intercepts the flow between two nodes $x$ and $y$ if $S$ intersect every shortest path between $x$ and $y$. Differently from what was conjectured by Jonckheere et al., we show that $m$ is not (and cannot be) the center of mass of $X$ but is a node close to the median of $X$ in the so-called injective hull of $X$. In case of non-uniform traffic between nodes of $X$ (in this case, the unit flow exists only between certain pairs of nodes of $X$ defined by a commodity graph $R$), we prove a primal-dual result showing that for any $\rho>5\delta$ the size of a $\rho$-multi-core (i.e., the number of disks of radius $\rho$) intercepting all pairs of $R$ is upper bounded by the maximum number of pairwise $(\rho-3\delta)$-apart pairs of $R$

Quantitative Tverberg, Helly, & Carath\'eodory theorems

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

A sunflower is a collection of sets $\{U_1,\ldots, U_n\}$ such that the pairwise intersection $U_i\cap U_j$ is the same for all choices of distinct $i$ and $j$. We study sunflowers of convex open sets in $\mathbb R^d$, and provide a Helly-type theorem describing a certain "rigidity" that they possess. In particular we show that if $\{U_1,\ldots, U_{d+1}\}$ is a sunflower in $\mathbb R^d$, then any hyperplane that intersects all $U_i$ must also intersect $\bigcap_{i=1}^{d+1} U_i$. We use our results to describe a combinatorial code $\mathcal C_n$ for all $n\ge 2$ 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 $\mathbf P_{\mathbf{Code}}$