Low-Complexity Decompositions of Combinatorial Objects
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Pseudorandom Ramsey Graphs
Pseudorandom Ramsey Graphs Jacques Verstraete, UCSD [email protected] Abstract We outline background to a recent theorem [47] connecting optimal pseu- dorandom graphs to the classical off-diagonal Ramsey numbers r(s; t) and graph Ramsey numbers r(F; t). A selection of exercises is provided. Contents 1 Linear Algebra 1 1.1 (n; d; λ)-graphs . 2 1.2 Alon-Boppana Theorem . 2 1.3 Expander Mixing Lemma . 2 2 Extremal (n; d; λ)-graphs 3 2.1 Triangle-free (n; d; λ)-graphs . 3 2.2 Clique-free (n; d; λ)-graphs . 3 2.3 Quadrilateral-free graphs . 4 3 Pseudorandom Ramsey graphs 4 3.1 Independent sets . 4 3.2 Counting independent sets . 5 3.3 Main Theorem . 5 4 Constructing Ramsey graphs 5 4.1 Subdivisions and random blocks . 6 4.2 Explicit Off-Diagonal Ramsey graphs . 7 5 Multicolor Ramsey 7 5.1 Blowups . 8 1. Linear Algebra X. Since trace is independent of basis, for k ≥ 1: n If A is a square symmetric matrix, then the eigenvalues X tr(Ak) = tr(X−kAkXk) = tr(Λk) = λk: (2) of A are real. When A is an n by n matrix, we denote i i=1 them λ1 ≥ λ2 ≥ · · · ≥ λn. If the corresponding eigen- k vectors forming an orthonormal basis are e1; e2; : : : ; en, The combinatorial interpretation of tr(A ) as the num- n P then for any x 2 R , we may write x = xiei and ber of closed walks of length k in the graph G will be used frequently. When A is the adjacency matrix of a n n X 2 X graph G, we write λi(G) for the ith largest eigenvalue hAx; xi = λixi and hAx; yi = λixiyi: (1) i=1 i=1 of A and For any i 2 [n], note xi = hx; eii. -
Pseudorandom Graphs
Pseudorandom graphs David Conlon What is a pseudorandom graph? A pseudorandom graph is a graph that behaves like a random graph of the same edge density. For example, it might be the case that the graph has roughly the same edge density between any two large sets or that it contains roughly the same number of copies of every small graph H as one would expect to find in a random graph. The purpose of this course will be to explore certain notions of pseudorandomness and to see what properties the resulting graphs have. For much of the course, our basic object of study will be (p; β)-jumbled graphs, a concept introduced (in a very slightly different form) by Andrew Thomason in the 1980s. These are defined as follows. Definition 1 A graph G on vertex set V is (p; β)-jumbled if, for all vertex subsets X; Y ⊆ V (G), je(X; Y ) − pjXjjY jj ≤ βpjXjjY j: That is, the edge density between any two sets X and Y is roughly p, with the allowed discrepancy measured in terms of β. The normalisation may be explained by considering what happens in a random graph. In this case, the expected number of edges between X and Y will be pjXjjY j and the standard deviation will be pp(1 − p)jXjjY j. So β is measuring how many multiples of the standard deviation our edge density is allowed to deviate by. Of course, random graphs should themselves be pseudorandom. Recall that the binomial random graph G(n; p) is a graph on n vertices formed by choosing each edge independently with probability p. -
The Regularity Method for Graphs with Few 4-Cycles
THE REGULARITY METHOD FOR GRAPHS WITH FEW 4-CYCLES DAVID CONLON, JACOB FOX, BENNY SUDAKOV, AND YUFEI ZHAO Abstract. We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: • Every n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n3=2) edges. • For r ≥ 3, every n-vertex r-graph with girth greater than 5 has o(n3=2) edges. • Every subset of [n] without a nontrivial solution to the equation x + x + 2x = x + 3x has p 1 2 3 4 5 size o( n). 1. Introduction Szemerédi’s regularity lemma [52] is a rough structure theorem that applies to all graphs. The lemma originated in Szemerédi’s proof of his celebrated theorem that dense sets of integers contain arbitrarily long arithmetic progressions [51] and is now considered one of the most useful and important results in combinatorics. Among its many applications, one of the earliest was the influential triangle removal lemma of Ruzsa and Szemerédi [40], which says that any n-vertex graph with o(n3) triangles can be made triangle-free by removing o(n2) edges. Surprisingly, this simple sounding statement is already sufficient to imply Roth’s theorem, the special case of Szemerédi’s theorem for 3-term arithmetic progressions, and a generalization known as the corners theorem. Most applications of the regularity lemma, including the triangle removal lemma, rely on also having an associated counting lemma. Such a lemma roughly says that the number of embeddings of a fixed graph H into a pseudorandom graph G can be estimated by pretending that G is a random graph. -
Arxiv:2104.11626V1 [Math.CO] 23 Apr 2021 Φ San Is Partition Ento 1.4 Definition Result
REMOVAL LEMMAS AND APPROXIMATE HOMOMORPHISMS JACOB FOX AND YUFEI ZHAO Abstract. We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each ǫ> 0 there exists M such that every triangle-free graph G has an ǫ-approximate homomorphism to a triangle- free graph F on at most M vertices (here an ǫ-approximate homomorphism is a map V (G) → V (F ) where all but at most ǫ |V (G)|2 edges of G are mapped to edges of F ). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any − polynomial in ǫ 1. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal. 1. Introduction 1.1. Graph removal and related results. The triangle removal lemma of Ruzsa and Sze- mer´edi [27] is a fundamental tool in extremal combinatorics. Theorem 1.1 (Triangle removal lemma). For every ǫ > 0, there exists δ > 0 such that every n-vertex graph with fewer than δn3 triangles can be made triangle-free by deleting at most ǫn2 edges. Definition 1.2. Let δT RL(ǫ) denote the largest possible constant δ in Theorem 1.1. The standard proof of the triangle removal lemma, which uses Szemer´edi’s regularity lemma [30], −1 −O(1) gives an upper bound on δT RL(ǫ) which is a tower of 2’s of height ǫ . -
On the Resilience of Hamiltonicity and Optimal Packing of Hamilton Cycles in Random Graphs*
SIAM J. DISCRETE MATH. © 2011 Society for Industrial and Applied Mathematics Vol. 25, No. 3, pp. 1176–1193 ON THE RESILIENCE OF HAMILTONICITY AND OPTIMAL PACKING OF HAMILTON CYCLES IN RANDOM GRAPHS* † ‡ § SONNY BEN-SHIMON , MICHAEL KRIVELEVICH , AND BENNY SUDAKOV Abstract. Let k ¼ðk1; :::;knÞ be a sequence of n integers. For an increasing monotone graph property P we say that a base graph G ¼ð½n;EÞ is k-resilient with respect to P if for every subgraph H ⊆ G such that dH ðiÞ ≤ ki for every 1 ≤ i ≤ n the graph G − H possesses P. This notion naturally extends the idea of the local resilienceofgraphsrecentlyinitiatedbySudakovandVu.Inthispaperwestudythek-resilienceofatypicalgraph from Gðn; pÞ with respect to the Hamiltonicity property, where we let p range over all values for which the base graph is expected to be Hamiltonian. Considering this generalized approach to the notion of resilience our main result implies several corollaries which improve on the best known bounds of Hamiltonicity related questions. For ε K ln n one, it implies that for every positive > 0 and large enough values of K,ifp> n , then with high probability the local resilience of Gðn; pÞ with respect to being Hamiltonian is at least ð1 − εÞnp∕ 3, improving on the previous bound for this range of p. Another implication is a result on optimal packing of edge-disjoint Hamilton cycles in a ≤ 1.02 ln n G random graph. We prove that if p n , then with high probability a graph G sampled from ðn; pÞ contains δðGÞ b 2 c edge-disjoint Hamilton cycles, extending the previous range of p for which this was known to hold. -
Finding Any Given 2‐Factor in Sparse Pseudorandom Graphs Efficiently
Received: 16 February 2019 | Revised: 7 April 2020 | Accepted: 18 April 2020 DOI: 10.1002/jgt.22576 ARTICLE Finding any given 2‐factor in sparse pseudorandom graphs efficiently Jie Han1 | Yoshiharu Kohayakawa2 | Patrick Morris3,4 | Yury Person5 1Department of Mathematics, University of Rhode Island, Kingston, Rhode Island Abstract 2Instituto de Matemáticae Estatística, Given an n‐vertex pseudorandom graph G and an Universidade de São Paulo, São Paulo, n‐vertex graph H with maximum degree at most two, Brazil we wish to find a copy of H in G, that is, an em- 3Institut für Mathematik, Freie Universität Berlin, Berlin, Germany bedding φ :()VH→ VG () so that φ()u φ ()vEG∈ ( ) 4Berlin Mathematical School, Berlin, for all uv∈ E() H . Particular instances of this pro- Germany blem include finding a triangle‐factor and finding a 5 Institut für Mathematik, Technische Hamilton cycle in G. Here, we provide a deterministic Universität, Ilmenau, Germany polynomial time algorithm that finds a given H in Correspondence any suitably pseudorandom graph G. The pseudor- Jie Han, Department of Mathematics, andom graphs we consider are (p,)λ ‐bijumbled University of Rhode Island, 5 Lippitt Road, Kingston, RI 02881. graphs of minimum degree which is a constant pro- Email: [email protected] portion of the average degree, that is, Ω()pn .A(p,)λ ‐ bijumbled graph is characterised through the dis- Funding information Fundação de Amparo à Pesquisa do Estado crepancy property: |eAB(, )− pA|||||< B λ ||||AB de São Paulo, Grant/Award Numbers: for any two sets of vertices A and B. Our condition 2013/03447‐6, 2014/18641‐5; Conselho λ =(Opn2 /log n) on bijumbledness is within a log Nacional de Desenvolvimento Científico e Tecnológico, Grant/Award Numbers: factor from being tight and provides a positive answer 310974/2013‐5, 311412/2018‐1, 423833/ to a recent question of Nenadov. -
MITOCW | 11. Pseudorandom Graphs I: Quasirandomness
MITOCW | 11. Pseudorandom graphs I: quasirandomness PROFESSOR: So we spent the last few lectures discussing Szemerédi's regularity lemma. So we saw that this is an important tool with important applications, allowing you to do things like a proof of Roth's theorem via graph theory. One of the concepts that came up when we were discussing the statement of Szemerédi's regularity lemma is that of pseudorandomness. So the statement of Szemerédi's graph regularity lemma is that you can partition an arbitrary graph into a bounded number of pieces so that the graph looks random-like, as we called it, between most pairs of parts. So what does random-like mean? So that's something that I want to discuss for the next couple of lectures. And this is the idea of pseudorandomness, which is a concept that is really prevalent in combinatorics, in theoretical computer science, and in many different areas. And what pseudorandomness tries to capture is, in what ways can a non-random object look random? So before diving into some specific mathematics, I want to offer some philosophical remarks. So you might know that, on a computer, you want to generate a random number. Well, you type in a "rand," and it gives you a random number. But of course, that's not necessarily true randomness. It came from some pseudorandom generator. Probably there's some seed and some complex-looking function and outputs something that you couldn't distinguish from random. But it might not actually be random but just something that looks, in many different ways, like random. -
Large Networks and Graph Limits
American Mathematical Society Colloquium Publications Volume 60 Large Networks and Graph Limits László Lovász Large Networks and Graph Limits http://dx.doi.org/10.1090/coll/060 American Mathematical Society Colloquium Publications Volume 60 Large Networks and Graph Limits László Lovász American Mathematical Society Providence, Rhode Island Editorial Board Lawrence C. Evans Yuri Manin Peter Sarnak (Chair) 2010 Mathematics Subject Classification. Primary 58J35, 58D17, 58B25, 19L64, 81R60, 19K56, 22E67, 32L25, 46L80, 17B69. For additional information and updates on this book, visit www.ams.org/bookpages/coll-60 ISBN-13: 978-0-8218-9085-1 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 To Kati as all my books Contents Preface xi Part 1. -
Arxiv:2004.10180V2 [Math.CO]
THE REGULARITY METHOD FOR GRAPHS WITH FEW 4-CYCLES DAVID CONLON, JACOB FOX, BENNY SUDAKOV, AND YUFEI ZHAO Abstract. We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: Every n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n3/2) edges. • For r 3, every n-vertex r-graph with girth greater than 5 has o(n3/2) edges. • ≥ Every subset of [n] without a nontrivial solution to the equation x1 + x2 + 2x3 = x4 + 3x5 has • size o(√n). 1. Introduction Szemerédi’s regularity lemma [52] is a rough structure theorem that applies to all graphs. The lemma originated in Szemerédi’s proof of his celebrated theorem that dense sets of integers contain arbitrarily long arithmetic progressions [51] and is now considered one of the most useful and important results in combinatorics. Among its many applications, one of the earliest was the influential triangle removal lemma of Ruzsa and Szemerédi [40], which says that any n-vertex graph with o(n3) triangles can be made triangle-free by removing o(n2) edges. Surprisingly, this simple sounding statement is already sufficient to imply Roth’s theorem, the special case of Szemerédi’s theorem for 3-term arithmetic progressions, and a generalization known as the corners theorem. Most applications of the regularity lemma, including the triangle removal lemma, rely on also having an associated counting lemma. Such a lemma roughly says that the number of embeddings of a fixed graph H into a pseudorandom graph G can be estimated by pretending that G is a random graph. -
Arxiv:1402.0984V2
POWERS OF HAMILTON CYCLES IN PSEUDORANDOM GRAPHS PETER ALLEN, JULIA BOTTCHER,¨ HIEˆ. P HAN,` YOSHIHARU KOHAYAKAWA, AND YURY PERSON Abstract. We study the appearance of powers of Hamilton cycles in pseudo- random graphs, using the following comparatively weak pseudorandomness no- tion. A graph G is (ε,p,k,ℓ)-pseudorandom if for all disjoint X and Y ⊆ V (G) with |X|≥ εpkn and |Y |≥ εpℓn we have e(X,Y ) = (1 ± ε)p|X||Y |. We prove that for all β > 0 there is an ε> 0 such that an (ε,p, 1, 2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamil- ton cycle. In particular, this implies that (n, d, λ)-graphs with λ ≪ d5/2n−3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szab´o [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions. 1. Introduction and results The appearance of certain graphs H as subgraphs is a dominant topic in the study of random graphs. In the random graph model G(n,p) this question turned out to be comparatively easy for graphs H of constant size, but much harder for graphs H on n vertices, i.e., spanning subgraphs. Early results were however obtained in the case when H is a Hamilton cycle, for which this question is by now very well understood [8, 19, 20, 21, 27]. -
Szemerédi's Regularity Lemma for Sparse Graphs
Szemer´edi'sRegularity Lemma for Sparse Graphs Y. Kohayakawa? Instituto de Matem´aticae Estat´ıstica,Universidade de S~aoPaulo Rua do Mat~ao1010, 05508{900 S~aoPaulo, SP Brazil Abstract. A remarkable lemma of Szemer´ediasserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This far-reaching result has proved to play a central r^olein many areas of combinatorics, both `pure' and `algorithmic.' The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far. 1 Introduction Szemer´edi'scelebrated proof [39] of the conjecture of Erd}osand Tur´an[10] on arithmetic progressions in dense subsets of integers is certainly a masterpiece of modern combinatorics. An auxiliary lemma in that work, which has become known in its full generality [40] as Szemer´edi'sregularity lemma, has turned out to be a powerful and widely applicable combinatorial tool. For an authoritative survey on this subject, the reader is referred to the recent paper of Koml´osand Simonovits [29]. For the algorithmic aspects of this lemma, the reader is referred to the papers of Alon, Duke, Lefmann, R¨odl,and Yuster [1] and Duke, Lefmann, and R¨odl[8]. Very roughly speaking, the lemma of Szemer´edisays that any graph can be decomposed into a bounded number of pseudorandom bipartite graphs. Since pseudorandom graphs have a predictable structure, the regularity lemma is a powerful tool for introducing `order' where none is visible at first. -
Graph Theory and Additive Combinatorics, a Graduate-Level Course Taught by Prof
GRAPHTHEORY AND ADDITIVECOMBINATORICS notes for mit 18.217 (fall 2019) lecturer: yufei zhao http://yufeizhao.com/gtac/ About this document This document contains the course notes for Graph Theory and Additive Combinatorics, a graduate-level course taught by Prof. Yufei Zhao at MIT in Fall 2019. The notes were written by the students of the class based on the lectures, and edited with the help of the professor. The notes have not been thoroughly checked for accuracy, espe- cially attributions of results. They are intended to serve as study resources and not as a substitute for professionally prepared publica- tions. We apologize for any inadvertent inaccuracies or misrepresen- tations. More information about the course, including problem sets and lecture videos (to appear), can be found on the course website: http://yufeizhao.com/gtac/ Contents A guide to editing this document 7 1 Introduction 13 1.1 Schur’s theorem........................ 13 1.2 Highlights from additive combinatorics.......... 15 1.3 What’s next?.......................... 18 I Graph theory 21 2 Forbidding a subgraph 23 2.1 Mantel’s theorem: forbidding a triangle.......... 23 2.2 Turán’s theorem: forbidding a clique............ 24 2.3 Hypergraph Turán problem................. 26 2.4 Erd˝os–Stone–Simonovits theorem (statement): forbidding a general subgraph...................... 27 2.5 K˝ovári–Sós–Turán theorem: forbidding a complete bipar- tite graph............................ 28 2.6 Lower bounds: randomized constructions......... 31 2.7 Lower bounds: algebraic constructions.......... 34 2.8 Lower bounds: randomized algebraic constructions... 37 2.9 Forbidding a sparse bipartite graph............ 40 3 Szemerédi’s regularity lemma 49 3.1 Statement and proof....................