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 ℝ ^d, 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 {H _n} ^∞ _n=1 of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree for which inf c{H _n) > 0. Using both random methods and explicit constructions, n≧1 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 h; s and any ε > 0, there exists K - K(ε; h; s) satisfying the following condition. For any k ≧ K and for any semi-algebraic relation R on h-tuples of points in a Euclidean space ℝ ^d with description complexity at most s, every finite set P ⊆ ℝ ^d has a partition P - P1 ∪⋯ ∪ P _k into k parts of sizes as equal as possible such that all but at most an ε-fraction of the h-tuples (P _i1;⋯; P _ih) have the property that either all h-tuples of points with one element in each P _ij are related with respect to R or none of them are.