# Overlap properties of geometric expanders

@inproceedings{Fox2010OverlapPO, 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}, year={2010} }

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…

## 118 Citations

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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 $K_p$.

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