### 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 ℝ
^{d} there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. In [18], motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence {H
_{n}}
_{n=1}
^{∞} of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree, for which inf
_{n≥1} c(H
_{n}) > 0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any d and any ε > 0, there exists K = K(ε,d) ≥ d + 1 satisfying the following condition. For any k ≥ K, for any point q ∈ ℝ
^{d} and for any finite Borei measure μ on ℝ
^{d} with respect to which every hyperplane has measure 0, there is a partition ℝ
^{d} = A
_{1} ∪. .. ∪ A
_{k} into k measurable parts of equal measure such that all but at most an ε-fraction of the (d + 1)-tuples A
_{i1},..., A
_{id+1} have the property that either all simplices with one vertex in each A
_{ij} contain q or none of these simplices contain q.

Original language | English (US) |
---|---|

Title of host publication | Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 |

Pages | 1188-1197 |

Number of pages | 10 |

State | Published - 2011 |

Event | 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 - San Francisco, CA, United States Duration: Jan 23 2011 → Jan 25 2011 |

### Other

Other | 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 |
---|---|

Country | United States |

City | San Francisco, CA |

Period | 1/23/11 → 1/25/11 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011*(pp. 1188-1197)

**Overlap properties of geometric expanders.** / Fox, Jacob; Gromov, Mikhael; Lafforgue, Vincent; Naor, Assaf; Pach, János.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011.*pp. 1188-1197, 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, CA, United States, 1/23/11.

}

TY - GEN

T1 - Overlap properties of geometric expanders

AU - Fox, Jacob

AU - Gromov, Mikhael

AU - Lafforgue, Vincent

AU - Naor, Assaf

AU - Pach, János

PY - 2011

Y1 - 2011

N2 - 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 ℝ d there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. In [18], motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence {H n} n=1 ∞ of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree, for which inf n≥1 c(H n) > 0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any d and any ε > 0, there exists K = K(ε,d) ≥ d + 1 satisfying the following condition. For any k ≥ K, for any point q ∈ ℝ d and for any finite Borei measure μ on ℝ d with respect to which every hyperplane has measure 0, there is a partition ℝ d = A 1 ∪. .. ∪ A k into k measurable parts of equal measure such that all but at most an ε-fraction of the (d + 1)-tuples A i1,..., A id+1 have the property that either all simplices with one vertex in each A ij contain q or none of these simplices contain q.

AB - 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 ℝ d there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. In [18], motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence {H n} n=1 ∞ of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree, for which inf n≥1 c(H n) > 0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any d and any ε > 0, there exists K = K(ε,d) ≥ d + 1 satisfying the following condition. For any k ≥ K, for any point q ∈ ℝ d and for any finite Borei measure μ on ℝ d with respect to which every hyperplane has measure 0, there is a partition ℝ d = A 1 ∪. .. ∪ A k into k measurable parts of equal measure such that all but at most an ε-fraction of the (d + 1)-tuples A i1,..., A id+1 have the property that either all simplices with one vertex in each A ij contain q or none of these simplices contain q.

UR - http://www.scopus.com/inward/record.url?scp=79955736052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79955736052&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:79955736052

SN - 9780898719932

SP - 1188

EP - 1197

BT - Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011

ER -