Graph Theory and Additive Combinatorics, a Graduate-Level Course Taught by Prof
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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. -
Graph Theory
Graph theory Eric Shen Friday, August 28, 2020 The contents of this handout are based on various lectures on graph theory I have heard over the years, including some from MOP and some from Canada/USA Mathcamp. Contents 1. Basic terminology3 1.1. Introduction to graphs ................................ 3 1.2. Dictionary ....................................... 3 2. Basic graph facts4 2.1. Trees .......................................... 4 2.2. Handshaking lemma.................................. 5 2.3. Example problems................................... 5 3. Colorings of graphs7 3.1. Introduction to coloring................................ 7 3.2. Planar graphs ..................................... 7 3.3. Four-color theorem .................................. 8 3.4. Example problems................................... 8 4. Combinatorial constructions 10 4.1. Basic graph constructions............................... 10 4.2. Perfect matchings................................... 11 4.3. Corollaries of Hall's theorem............................. 11 4.4. A hard walkthrough.................................. 12 5. Extremal graph theory 13 5.1. Triangle-free bounds.................................. 13 5.2. Formalization ..................................... 14 5.3. Tur´angraphs...................................... 14 5.4. Example problems................................... 15 6. Application: crossing numbers 17 6.1. Conventional definitions................................ 17 6.2. Some conjectures ................................... 17 6.3. Crossing number -
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 . -
The Chromatic Number of Infinite Graphs —
Discrete Mathematics 311 (2011) 1448–1450 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc The chromatic number of infinite graphs — A survey Péter Komjáth Department of Computer Science, Eötvös University, Budapest, P.O. Box 120, 1518, Hungary article info a b s t r a c t Article history: We survey some old and new results on the chromatic number of infinite graphs. Available online 26 November 2010 ' 2010 Elsevier B.V. All rights reserved. Keywords: Infinite graphs Chromatic number 1. Coloring number The coloring number of a graph X; Col.X/, is the smallest cardinal µ such that V , the set of vertices of X, has a well- ordering < such that each vertex is joined into less than µ smaller (by <) vertices. This notion was introduced by Erd®s and Hajnal in [2]. If Col.X/ D κ and < is a well-ordering witnessing it, then a straightforward transfinite recursion via < gives a good coloring of X with κ colors; at each step we can continue the coloring. This establishes Chr.X/ ≤ Col.X/ for any graph C X. Equality may not hold even for very simple graphs as Col.Kκ,κ / D κ and Kκ,λ D κ for κ < λ. Here Kκ,λ denotes the complete bipartite graph with bipartition classes of size κ; λ, respectively. What makes the notion of coloring number particularly useful is Shelah's Singular Cardinals Compactness Theorem: if λ > µ, λ is singular, X is a graph such that Col.Y / ≤ µ holds for every subgraph Y of X with jY j < λ, then Col.X/ ≤ µ holds, as well [22]. -
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. -
Bounds for Graph Regularity and Removal Lemmas
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace@MIT Bounds for graph regularity and removal lemmas David Conlon∗ Jacob Foxy Abstract We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck2= log∗ k pairs of parts which are not -regular, where c; > 0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer´edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter may require as many as −2 2Ω( ) parts. This is tight up to the implied constant and solves a problem studied by Lov´asz and Szegedy. 1 Introduction Originally developed by Szemer´edias part of his proof of the celebrated Erd}os-Tur´anconjecture on long arithmetic progressions in dense subsets of the integers [39], Szemer´edi'sregularity lemma [40] has become a central tool in extremal combinatorics. -
Large Networks and Graph Limits László Lovász
Large networks and graph limits L´aszl´oLov´asz Institute of Mathematics, Eotv¨ os¨ Lorand´ University, Budapest, Hungary 2010 Mathematics Subject Classification. Primary 05C99, Secondary 05C25, 05C35, 05C80, 05C82, 05C85, 90B15 Key words and phrases. graph homomorphism, graph algebra, graph limit, graphon, graphing, property testing, regularity lemma To Kati as all my books Contents Preface xi Part 1. Large graphs: an informal introduction 1 Chapter 1. Very large networks 3 1.1. Huge networks everywhere 3 1.2. What to ask about them? 4 1.3. How to obtain information about them? 5 1.4. How to model them? 8 1.5. How to approximate them? 11 1.6. How to run algorithms on them? 18 1.7. Bounded degree graphs 22 Chapter 2. Large graphs in mathematics and physics 25 2.1. Extremal graph theory 25 2.2. Statistical physics 32 Part 2. The algebra of graph homomorphisms 35 Chapter 3. Notation and terminology 37 3.1. Basic notation 37 3.2. Graph theory 38 3.3. Operations on graphs 39 Chapter 4. Graph parameters and connection matrices 41 4.1. Graph parameters and graph properties 41 4.2. Connection matrices 42 4.3. Finite connection rank 45 Chapter 5. Graph homomorphisms 55 5.1. Existence of homomorphisms 55 5.2. Homomorphism numbers 56 5.3. What hom functions can express 62 5.4. Homomorphism and isomorphism 68 5.5. Independence of homomorphism functions 72 5.6. Characterizing homomorphism numbers 75 5.7. The structure of the homomorphism set 79 Chapter 6. Graph algebras and homomorphism functions 83 6.1. -
Regular Partitions of Gentle Graphs
REGULAR PARTITIONS OF GENTLE GRAPHS YITING JIANG, JAROSLAV NESETˇ RIL,ˇ PATRICE OSSONA DE MENDEZ, AND SEBASTIAN SIEBERTZ Dedicated to Endre Szemer´edion the occasion of his eightieth birthday. Abstract. Szemer´edi'sRegularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as \gentle" classes. Contents Introduction2 Part 1. Preliminaries and survey of some regularity lemmas 4 1. Model theory background4 2. Graph theoretic background5 3. Random-free regularity lemmas9 Part 2. Regularity for gentle graphs 15 4. Set-defined classes: both stable and semi-algebraic regularity 15 5. Order-defined classes 18 6. Inherited regularity of 2-covered classes 19 7. Regularity for classes 2-covered by embedded m-partite cographs 22 8. Regularity and non-regularity for nowhere dense classes 26 Conclusion 29 arXiv:2003.11692v2 [math.CO] 29 Mar 2020 Acknowledgments 30 References 30 This paper is part of a project that has received funding from the European Research Council (ERC) under the Euro- pean Union's Horizon 2020 research and innovation programme (grant agreement No 810115 { Dynasnet). 1 REGULAR PARTITIONS OF GENTLE GRAPHS 2 Introduction Szemer´edi'sRegularity Lemma. Szemer´edi'sRegularity Lemma [77] is a very useful tool in extremal graph theory. -
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.................... -
[Cs.DM] 12 Aug 2021 Agnostic Online Learning and Excellent Sets
Agnostic Online Learning and Excellent Sets M. Malliaris∗ and S. Moran† September 6, 2021 Abstract We use algorithmic methods from online learning to revisit a key idea from the interaction of model theory and combinatorics, the existence of large “indivisible” sets, called “ǫ-excellent,” in k-edge stable graphs (equivalently, Littlestone classes). These sets arise in the Stable Regularity Lemma, a theorem characterizing the appearance of irregular pairs in Szemer´edi’s celebrated Regularity Lemma. Translating to the language of probability, we find a quite different existence proof for ǫ-excellent sets in Littlestone classes, using regret bounds in online learning. This proof k applies to any ǫ < 1/2, compared to < 1/22 or so in the original proof. We include a second proof using closure properties and the VC theorem, with other advantages but weaker bounds. As a simple corollary, the Littlestone dimension remains finite under some natural modifications to the definition. A theme in these proofs is the interaction of two abstract notions of majority, arising from measure, and from rank or dimension; we prove that these densely often coincide and that this is characteristic of Littlestone (stable) classes. The last section lists several open problems. 1 Overview In this section we briefly present three complementary points of view (combinatorics §1.1, online learning §1.2, model theory §1.3) which inform this work, and state our main results in §1.4. The aim is to allow the paper to be readable by people in all three communities and hopefully to stimulate further interaction. In the recent papers [4], [7] ideas from model theory played a role in the conjecture, and then arXiv:2108.05569v2 [cs.DM] 3 Sep 2021 the proof, that Littlestone classes are precisely those which can be PAC learned in a differentially private way (we direct the reader to the introductions of those papers for precise statements and further literature review). -
Low-Complexity Decompositions of Combinatorial Objects
IMPA Master's Thesis Low-Complexity Decompositions of Combinatorial Objects Author: Supervisor: Davi Castro-Silva Roberto Imbuzeiro Oliveira A thesis submitted in fulfillment of the requirements for the degree of Master in Mathematics at Instituto de Matem´aticaPura e Aplicada April 15, 2018 iii Contents 1 Introduction 1 1.1 High-level overview of the framework.........................1 1.2 Remarks on notation and terminology........................2 1.3 Examples and structure of the thesis.........................3 2 Abstract decomposition theorems7 2.1 Probabilistic setting..................................7 2.2 Structure and pseudorandomness...........................8 2.3 Weak decompositions..................................9 2.3.1 Weak Regularity Lemma........................... 10 2.4 Strong decompositions................................. 10 2.4.1 Szemer´ediRegularity Lemma......................... 11 3 Counting subgraphs and the Graph Removal Lemma 15 3.1 Counting subgraphs globally.............................. 15 3.2 Counting subgraphs locally and the Removal Lemma................ 17 3.3 Application to property testing............................ 19 4 Extensions of graph regularity 21 4.1 Regular approximation................................. 21 4.2 Relative regularity................................... 23 5 Hypergraph regularity 27 5.1 Intuition and definitions................................ 27 5.2 Regularity at a single level............................... 28 5.3 Regularizing all levels simultaneously........................