Overlap properties of geometric expanders

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Overlap properties of geometric expanders
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. In [17], 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 {Hn} n=1 of arbitrarily large (d+1)-uniform hypergraphs with bounded degree, for which infn 1 c(Hn) > 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 com...
Jacob Fox, Mikhail Gromov, Vincent Lafforgue, Assa
Added 09 Dec 2010
Updated 09 Dec 2010
Type Journal
Year 2010
Where CORR
Authors Jacob Fox, Mikhail Gromov, Vincent Lafforgue, Assaf Naor, János Pach
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