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...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Jacob Fox, Mikhail Gromov, Vincent Lafforgue, Assaf Naor, János Pach
• Keywords: -uniform Hypergraphs, CORR 2010, Dimensional Simplicial Complexes, Education, Overlap Numbers