Let H be a k-uniform hypergraph whose vertices are the integers 1, . . . , N . We say that H contains a monotone path of length n if there are x1 < x2 < Â· Â· Â· < xn+kâˆ’1 so that H contains all n edgesâ€¦ (More)

Let Wn(p, q) denote the minimum number of edges in an nÃ— n bipartite graph G on vertex sets X,Y that satisfies the following condition; one can add the edges between X and Y that do not belong to Gâ€¦ (More)

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemeredi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : Gâ€¦ (More)

Let Tr(n,m, k) denote the largest number of distinct projections onto k coordinates guaranteed in any family of m binary vectors of length n. The classical Sauer-Perles-Shelah Lemma implies thatâ€¦ (More)

We present a combinatorial approach for proving complexity lower bounds; we focus on the following instantiation of this approach. For a property of regular hypergraphs with m edges and an arbitraryâ€¦ (More)