Overlap properties of geometric expanders

  title={Overlap properties of geometric expanders},
  author={Jacob Fox and Misha Gromov and Vincent Lafforgue and Assaf Naor and J{\'a}nos Pach},
  booktitle={ACM-SIAM Symposium on Discrete Algorithms},
Abstract The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) ∈ (0, 1] such that no matter how we map the vertices of H into ℝd, there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. Motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, we address the question whether or not there exists a sequence of arbitrarily large (d + 1)-uniform… 

Density and regularity theorems for semi-algebraic hypergraphs

It is shown that for any typical hereditary hypergraph property Q, there is a randomized algorithm with query complexity to determine whether a k-uniform semi-algebraic hypergraph H = (P, E) with constant description complexity is e-near to having propertyQ, that is, whether one can change at most e|P|k hyperedges of H in order to obtain a hypergraph that has the property.

Local and Global Expansion in Random Geometric Graphs

Consider a random geometric 2-dimensional simplicial complex X sampled as follows: first, sample n vectors u1,…,un uniformly at random on Sd−1; then, for each triple i,j,k ∈ [n], add {i,j,k} and all

The Schur-Erdős problem for semi-algebraic colorings

We consider m-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case m = 2 was first studied by Alon et al., who applied

Random Steiner systems and bounded degree coboundary expanders of every dimension

A new model of random d-dimensional simplicial complexes, whose bounded degrees have bounded degrees is introduced, and it is shown that with high probability, complexes sampled according to this model are coboundary expanders.

C O ] 5 D ec 2 01 8 Semi-algebraic colorings of complete graphs

We consider m-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case m = 2 was first studied by Alon et al., who applied

Systolic Expanders of Every Dimension

This work shows explicit bounded degree systolic expanders of every dimension, which solves affirmatively an open question raised by Gromov [G], who asked whether there exists bounded degree simplicial complexes with the topological overlapping property in every dimension.

Geometric Intersection Patterns and the Theory of Topological Graphs 3 Using similar ideas

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown

Independent sets in algebraic hypergraphs

In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, in the spirit of the so-called transference principles in extremal combinatorics, that if a given

Semi-algebraic colorings of complete graphs

This work considers-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity, and proves that the classical Ramsey number R(p;m) is the smallest positive integer such that any $m$-coloring of the edge of the complete graph on $n$ vertices, contains a monochromatic $K_p$.

Bounded VC-Dimension Implies the Schur-Erdős Conjecture

The conjecture for $m$-colorings with bounded VC-dimension is proved, that is, for £m- Colorings with the property that the set system $\mathcal{F}$ induced by the neighborhoods of the vertices with respect to each color class has boundedVC-dimension.

Stabbing Simplices by Points and Flats

An equipartition result of independent interest is established (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in ℝd can be divided into 4d−2 equal parts by 2d−1 hyperplanes intersecting in a common ( d−2)-flat.

The Colored Tverberg's Problem and Complexes of Injective Functions

A Tverberg-type result on multicolored simplices

  • J. Pach
  • Mathematics
    Comput. Geom.
  • 1998


We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the

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Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication

Existence of discrete cocompact subgroups of reductive groups over local fields.

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Eppstein's bound on intersecting triangles revisited