arXiv subjects

Guy Moshkovitz

Publications and source records attributed to Guy Moshkovitz.

22 records · Page 2Linked to original sources

arXiv2014

An Improved Lower Bound for Arithmetic Regularity

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{é}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup $H \le G$ 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 small. Quantitatively, if one wishes to obtain that for $1-ε$ fraction of the cosets, the nontrivial Fourier coefficients are bounded by $ε$, then Green shows that $|G/H|$ is bounded by a tower of twos of height $1/ε^3$. He also gives an example showing that a tower of height $Ω(\log 1/ε)$ is necessary. Here, we give an improved example, showing that a tower of height $Ω(1/ε)$ is necessary.

math.CO↗

arXiv2014

Exact Bounds for Some Hypergraph Saturation Problems

Let W_n(p,q) denote the minimum number of edges in an n x 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 G one after the other so that whenever a new edge is added, a new copy of K_{p,q} is created. The problem of bounding W_n(p,q), and its natural hypergraph generalization, was introduced by Balogh, Bollobás, Morris and Riordan. Their main result, specialized to graphs, used algebraic methods to determine W_n(1,q). Our main results in this paper give exact bounds for W_n(p,q), its hypergraph analogue, as well as for a new variant of Bollobás's Two Families theorem. In particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n then W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2. 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.

math.CO↗

arXiv2013

A Short Proof of Gowers' Lower Bound for the Regularity Lemma

A celebrated result of Gowers states that for every ε> 0 there is a graph G so that every ε-regular partition of G (in the sense of Szemeredi's regularity lemma) has order given by a tower of exponents of height polynomial in 1/ε. 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.

math.CO↗

arXiv2012

Ramsey Theory, Integer Partitions and a New Proof of the Erdos-Szekeres Theorem

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 x_1 < x_2 < ... < x_{n+k-1} so that H contains all n edges of the form {x_i,x_{i+1},...,x_{i+k-1}}. Let N_k(q,n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of N_k(q,n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdos and Szekeres, the problem of bounding N_k(q,n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk. Our main contribution here is a novel approach for bounding the Ramsey-type numbers N_k(q,n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following: 1. We show that for every fixed q we have N_3(q,n)=2^{Θ(n^{q-1})}, thus resolving an open problem raised by Fox et al. 2. We show that for every k >= 3, N_k(2,n)=2^{\cdot^{\cdot^{2^{(2-o(1))n}}}} where the height of the tower is k-2, thus resolving an open problem raised by Elias and Matousek. 3. We give a new pigeonhole proof of the Erdős-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdos-Szekeres Lemma on increasing/decreasing subsequences.

math.CO↗