Guy Moshkovitz

A short proof of Gowers' lower bound for the regularity lemma

A celebrated result of Gowers states that for every є>0 there is a graph G such that every є-regular partition of G (in the sense of Szemerédi’s regularity lemma) has order given by a tower of exponents of height polynomial in 1/є. In this note we give a new proof of this result that uses a construction and proof of correctness that are significantly simpler and shorter.

Exact bounds for some hypergraph saturation problems

Let K p 1 , ? , p d d denote the complete d-uniform d-partite hypergraph with partition classes of sizes p 1 , ? , p d . A hypergraph G ? K n , ? , n d is said to be weakly K p 1 , ? , p d d -saturated if one can add the edges of K n , ? , n d ? G in some order so that at each step a new copy of K p 1 , ? , p d d is created. Let W n ( p 1 , ? , p d ) denote the minimum number of edges in such a hypergraph. The problem of bounding W n ( p 1 , ? , p d ) was introduced by Balogh, Bollobas, Morris and Riordan who determined it when each p i is either 1 or some fixed p. In this note we fully determine W n ( p 1 , ? , p d ) . Our proof applies a reduction to a multi-partite version of the Two Families Theorem obtained by Alon. While the reduction is combinatorial, the main idea behind it is algebraic.

An Improved Lower Bound for Arithmetic Regularity

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 : G → (0,1), there exists a subgroup H ≤ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are s Quantitatively, if one wishes to obtain that for 1−ǫ fraction of the cosets, the nontrivial Fourier coefficients are bounded by ǫ, then Green shows that |G/H| is bounded by a tower of twos of height 1/ǫ 3 . He also gives an example showing that a tower of height (log1/ǫ) is necessary. Here, we give an improved example, showing that a tower of height (1/ǫ) is necessary.

Complexity Lower Bounds through Balanced Graph Properties

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 arbitrary hypergraph G with m - t edges, we may count the number of super-hypergraphs of G (i.e., hypergraphs obtained by adding t edges) satisfying the property. Suppose that we find a pair of these properties where every such G has the same number of super-hypergraphs satisfying each property. We show that in this case, we immediately obtain an explicit m/(t - 1) lower bound on the rank of tensors (which are high-dimensional matrices). Notice that if the hypergraphs are 3-uniform, this implies a lower bound of Ω(m/t) for arithmetic circuits. We also show, albeit non-explicitly, that essentially-optimal lower bounds can be obtained using this approach. Furthermore, we exemplify our approach in the t = 2 case, and prove that even in this case we can already obtain interesting lower bounds. In particular, we derive a (tight) lower bound of 3n/2 on the rank of n × n × n tensors that are naturally associated with hypergraph trees. In fact, our bound also applies to the stronger notion of border rank, for which our result essentially matches the best lower bounds known.