Regularity Lemma Research Articles

Erdős–Hajnal Conjecture for Graphs with Bounded VC-Dimension

The Vapnik–Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $$e^{(\log n)^{1 - o(1)}}$$ . The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, $$e^{c\sqrt{\log n}}$$ , due to Erdős and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erdős–Hajnal conjecture, according to which one can always find a clique or an independent set of size at least $$e^{\Omega (\log n)}$$ . Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz–Szegedy and Alon–Fischer–Newman, which is called the “ultra-strong regularity lemma” for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be $$(1/\varepsilon )^{O(d)}$$ , improving the original bound of $$(1/\varepsilon )^{O(d^2)}$$ in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an $$O(n^k)$$ -time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey–Turan numbers for graphs with bounded VC-dimension.

Definable sets containing productsets in expansions of groups

Abstract We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula x ⋅ y ∈ A {x\cdot y\in A} is stable. For arbitrary expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.

Lower bounds for the smallest singular value of structured random matrices

We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),\] where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemeredi’s regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.

Stable arithmetic regularity in the finite field model

The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in 2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$ of bounded codimension such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. It is known that in general, the growth of the codimension of $H$ is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non-uniform cosets. Our main result is that, under a natural model-theoretic assumption of stability, the tower-type bound and non-uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we prove an arithmetic regularity lemma for $k$-stable subsets $A\subseteq \mathbb{F}_p^n$ in which the bound on the codimension of the subspace is a polynomial (depending on $k$) in the degree of uniformity, and in which there are no non-uniform cosets. This result is an arithmetic analogue of the stable graph regularity lemma proved by Malliaris and Shelah.

The removal lemma for tournaments

Suppose one needs to change the direction of at least εn2 edges of an n-vertex tournament T, in order to make it H-free. A standard application of the regularity method shows that in this case T contains at least fH⁎(ε)nh copies of H, where fH⁎ is some tower-type function. It has long been observed that many graph/digraph problems become easier when assuming that the host graph is a tournament. It is thus natural to ask if the removal lemma becomes easier if we assume that the digraph G is a tournament.Our main result here is a precise characterization of the tournaments H for which fH⁎(ε) is polynomial in ε, stating that such a bound is attainable if and only if H's vertex set can be partitioned into two sets, each spanning an acyclic directed graph. The proof of this characterization relies, among other things, on a novel application of a regularity lemma for matrices due to Alon, Fischer and Newman, and on probabilistic variants of Ruzsa–Szemerédi graphs.We finally show that even when restricted to tournaments, deciding if H satisfies the condition of our characterization is an NP-hard problem.

Regularity lemma for distal structures

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces up to a small error (see e.g. [33, 2, 16, 18]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model-theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary o-minimal structures and in p -adics.

Erratum: On Regularity Lemmas and their Algorithmic Applications

Erratum: On Regularity Lemmas and their Algorithmic Applications

Constructive non-commutative rank computation is in deterministic polynomial time

We extend the techniques developed in Ivanyos et al. (Comput Complex 26(3):717–763, 2017) to obtain a deterministic polynomial-time algorithm for computing the non-commutative rank of linear spaces of matrices over any field. The key new idea that causes a reduction in the time complexity of the algorithm in Ivanyos et al. (2017) from exponential time to polynomial time is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen & Makam (Adv Math 310:44–63, 2017b), who were the first to observe that the blow-up parameter can be controlled. Both methods rely crucially on the regularity lemma from Ivanyos et al. (2017). In this note, we improve that lemma by removing a coprime condition there.

$L_p$ regular sparse hypergraphs

We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.

An algorithmic hypergraph regularity lemma

AbstractSzemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.

Unified mean-field framework for susceptible-infected-susceptible epidemics on networks, based on graph partitioning and the isoperimetric inequality.

We propose an approximation framework that unifies and generalizes a number of existing mean-field approximation methods for the susceptible-infected-susceptible (SIS) epidemic model on complex networks. We derive the framework, which we call the unified mean-field framework (UMFF), as a set of approximations of the exact Markovian SIS equations. Our main novelty is that we describe the mean-field approximations from the perspective of the isoperimetric problem, which results in bounds on the UMFF approximation error. These new bounds provide insight in the accuracy of existing mean-field methods, such as the N-intertwined mean-field approximation and heterogeneous mean-field method, which are contained by UMFF. Additionally, the isoperimetric inequality relates the UMFF approximation accuracy to the regularity notions of Szemerédi's regularity lemma.

Distributing pairs of vertices on Hamiltonian cycles

Let G be a graph of order n with minimum degree $$\delta (G) \geqslant \tfrac{n} {2} + 1$$ . Faudree and Li (2012) conjectured that for any pair of vertices x and y in G and any integer $$2 \leqslant k \leqslant \tfrac{n} {2}$$ , there exists a Hamiltonian cycle C such that the distance between x and y on C is k. In this paper, we prove that this conjecture is true for graphs of sufficiently large order. The main tools of our proof are the Regularity lemma of Szemeredi and the Blow-up lemma of Komlos et al. (1997).

Limits of discrete distributions and Gibbs measures on random graphs

Building upon the theory of graph limits and the Aldous–Hoover representation and inspired by Panchenko’s work on asymptotic Gibbs measures [Annals of Probability 2013], we construct continuous embeddings of discrete probability distributions. We show that the theory of graph limits induces a meaningful notion of convergence and derive a corresponding version of the Szemerédi regularity lemma. Moreover, complementing recent work Bapst et al. (2015), we apply these results to Gibbs measures induced by sparse random factor graphs and verify the “replica symmetric solution” predicted in the physics literature under the assumption of non-reconstruction.

Limits of structures and the example of tree semi-lattices

The notion of left convergent sequences of graphs introduced by Lovász et al. (in relation with homomorphism densities for fixed patterns and Szemerédi’s regularity lemma) got increasingly studied over the past 10 years. Recently, Nešetřil and Ossona de Mendez introduced a general framework for convergence of sequences of structures. In particular, the authors introduced the notion of QF-convergence, which is a natural generalization of left-convergence. In this paper, we initiate study of QF-convergence for structures with functional symbols by focusing on the particular case of tree semi-lattices. We fully characterize the limit objects and give an application to the study of left convergence of m-partite cographs, a generalization of cographs.

On non-commutative rank and tensor rank

We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give non-trivial equations for the tensors of border rank at most in if m is even. He also gave such equations for tensors of border rank at most in if m is odd. Using concavity of tensor blow-ups we show non-trivial equations for tensors of border rank in for odd m for any field of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam.

A tight lower bound for Szemerédi’s regularity lemma

We determine the order of the tower height for the partition size in a version of Szemeredi’s regularity lemma. This addresses a question of Gowers.

On Regularity Lemmas and their Algorithmic Applications

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

We prove that any 3-uniform hypergraph whose minimum vertex degree is at least 59+o(1)n2 admits an almost-spanning tight cycle, that is, a tight cycle leaving o(n) vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.

Polynomial degree bounds for matrix semi-invariants

We study the left–right action of SLn×SLn on m-tuples of n×n matrices with entries in an infinite field K. We show that invariants of degree n2−n define the null cone. Consequently, invariants of degree ≤n6 generate the ring of invariants if char(K)=0. We also prove that for m≫0, invariants of degree at least n⌊n+1⌋ are required to define the null cone. We generalize our results to matrix invariants of m-tuples of p×q matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.

Hereditary quasirandomness without regularity

AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.