The Probabilistic Method
|
|
- Clemence Garrison
- 5 years ago
- Views:
Transcription
1 The Probabilistic Method In Graph Theory Ehssan Khanmohammadi Department of Mathematics The Pennsylvania State University February 25, 2010
2 What do we mean by the probabilistic method? Why use this method? The Probabilistic Method and Paul Erdős The probabilistic method is a technique for proving the existence of an object with certain properties by showing that a random object chosen from an appropriate probability distribution has the desired properties with positive probability. Pioneered and championed by Paul Erdős who applied it mainly to problems in combinatorics and number theory from 1947 onwards.
3 What do we mean by the probabilistic method? Why use this method? An apocryphal story quoted from Molloy and Reed At every combinatorics conference attended by Erdős in 1960s and 1970s, there was at least one talk which concluded with Erdős informing the speaker that almost every graph was a counterexample to his/her conjecture!
4 What do we mean by the probabilistic method? Why use this method? Three facts about the probabilistic method which are worth bearing in mind: 1: Large and Unstructured Output Graphs The probabilistic method allows us to consider graphs which are both large and unstructured. The examples constructed using the probabilistic method routinely contain many, say 10 10, nodes. Explicit constructions necessarily introduce some structuredness to the class of graphs built, which thus restricts the graphs considered.
5 What do we mean by the probabilistic method? Why use this method? 2: Powerful and Easy to Use Erdős would routinely perform the necessary calculations to disprove a conjecture in his head during a fifteen-minute talk.
6 What do we mean by the probabilistic method? Why use this method? 2: Powerful and Easy to Use Erdős would routinely perform the necessary calculations to disprove a conjecture in his head during a fifteen-minute talk. 3: Covers almost Every Graph Erdős did not say some graph is a counterexample to your conjecture, but rather almost every graph is a counterexample to your conjecture.
7 Applications in Discrete Mathematics One can classify the applications of probabilistic techniques in discrete mathematics into two groups. 1: Study of Random Objects (Graphs, Matrices, etc.) A typical problem is the following: if we pick a graph at random, what is the probability that it contains a Hamiltonian cycle?
8 2: Proof of the Existence of Certain Structures Choose a structure randomly (from a probability distribution that you are free to specify). Estimate the probability that it has the properties you want. Show that this probability is greater than 0, and therefore conclude that such a structure exists.
9 2: Proof of the Existence of Certain Structures Choose a structure randomly (from a probability distribution that you are free to specify). Estimate the probability that it has the properties you want. Show that this probability is greater than 0, and therefore conclude that such a structure exists. Surprisingly often it is much easier to prove this than it is to give an example of a structure that works.
10 Example 1: Szele s result Two Definitions A directed complete graph is called a tournament. By a Hamiltonian path in a tournament we mean a path which traces each node (exactly once) following the direction of the graph.
11 Example 1: Szele s result Two Definitions A directed complete graph is called a tournament. By a Hamiltonian path in a tournament we mean a path which traces each node (exactly once) following the direction of the graph. The following result of Szele (1943) is ofttimes considered the first use of the probabilistic method. Theorem (Szele 1943) There is a tournament T with n players and at least n!2 (n 1) Hamiltonian paths.
12 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes.
13 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths.
14 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n.
15 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path.
16 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path. P(X σ = 1) = 2 (n 1).
17 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path. P(X σ = 1) = 2 (n 1). X = σ X σ, thus E(X ) = σ E(X σ) = n!2 (n 1).
18 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path. P(X σ = 1) = 2 (n 1). X = σ X σ, thus E(X ) = σ E(X σ) = n!2 (n 1). Conclusion: There is a tournament for which X is equal to at least E(X ).
19 Remark 1 A player who wins all games would naturally be the tournament s winner. However, there might not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament T = (V, E) is called k-paradoxical if for every k-element subset S of V there is a vertex v 0 in V \ S such that v 0 v for each v S. By means of the probabilistic method Erdős showed that, for any fixed value of k, if V is sufficiently large, then almost every tournament on V is k-paradoxical.
20 Remark 1 A player who wins all games would naturally be the tournament s winner. However, there might not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament T = (V, E) is called k-paradoxical if for every k-element subset S of V there is a vertex v 0 in V \ S such that v 0 v for each v S. By means of the probabilistic method Erdős showed that, for any fixed value of k, if V is sufficiently large, then almost every tournament on V is k-paradoxical. Remark 2 Szele conjectured that the maximum possible number of Hamiltonian paths in a tournament on n players is at most n! (2 o(1)) n. Alon proved this conjecture in 1990 using the probabilistic method.
21 Example 2: Lower bound for diagonal Ramsey numbers R(k, k) Definition The Ramsey number R(k, l) is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k or there is a blue K l.
22 Example 2: Lower bound for diagonal Ramsey numbers R(k, k) Definition The Ramsey number R(k, l) is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k or there is a blue K l. Ramsey (1929) showed that R(k, l) is finite for any two integers k, l.
23 Example 2: Lower bound for diagonal Ramsey numbers R(k, k) Definition The Ramsey number R(k, l) is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k or there is a blue K l. Ramsey (1929) showed that R(k, l) is finite for any two integers k, l. Theorem (Erdős (1947)) If ( n k) 2 1 ( k 2) < 1, then R(k, k) > n. Thus R(k, k) > 2 k/2 for each k 3.
24 Lower bound for R(k, k) (cont.) Proof. Consider a random 2-coloring of K n : Color each edge independently with probability 1 2 of being red and 1 2 of being blue.
25 Lower bound for R(k, k) (cont.) Proof. Consider a random 2-coloring of K n : Color each edge independently with probability 1 2 of being red and 1 2 of being blue. For any fixed set R of k nodes, let X R be the indicator of being monochromatic for induced subgraph of R, and define X for the whole graph similarly.
26 Lower bound for R(k, k) (cont.) Proof. Consider a random 2-coloring of K n : Color each edge independently with probability 1 2 of being red and 1 2 of being blue. For any fixed set R of k nodes, let X R be the indicator of being monochromatic for induced subgraph of R, and define X for the whole graph similarly. Clearly, P(X R = 1) = 2 1 (k 2), and by our assumption E(X ) = ( ) n E(X R ) = 2 1 (k 2) < 1 k R
27 Proof continued. E(X ) < 1, thus, there exists a complete graph on n nodes with no monochromatic subgraph on k nodes, because the expected value, that is, the mean number of monochromatic subgraphs is less than one, where the mean is taken over all 2-colorings of K n. So, R(k, k) > n.
28 Proof continued. E(X ) < 1, thus, there exists a complete graph on n nodes with no monochromatic subgraph on k nodes, because the expected value, that is, the mean number of monochromatic subgraphs is less than one, where the mean is taken over all 2-colorings of K n. So, R(k, k) > n. Note that if k 3 and we take n = 2 k/2, then ( ) n (k 2) 1+ k 2 < k k! and hence R(k, k) > 2 k/2. n k 2 k2 /2 < 1,
29 Example 3: A Result of Caro and Wei A Definition and a Notation A subset of the nodes of a graph is called independent if no two of its elements are adjacent. The size of a maximal (with respect to inclusion) independent set in a graph G = (V, E) is denoted by α(g).
30 Example 3: A Result of Caro and Wei A Definition and a Notation A subset of the nodes of a graph is called independent if no two of its elements are adjacent. The size of a maximal (with respect to inclusion) independent set in a graph G = (V, E) is denoted by α(g). Theorem (Caro (1979), Wei (1981)) α(g) v V 1 d v +1.
31 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }.
32 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }. Let X v be the indicator random variable for v I and X = v V X v = I.
33 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }. Let X v be the indicator random variable for v I and X = v V X v = I. For each v, E(X v ) = P(v I ) = 1 d v +1, since v I iff v is the least element among v and its neighbors.
34 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }. Let X v be the indicator random variable for v I and X = v V X v = I. For each v, E(X v ) = P(v I ) = 1 d v +1, since v I iff v is the least element among v and its neighbors. 1 d v +1 Hence E(X ) = v V, and so there exists a specific ordering < with I v V 1 d v +1.
35 Explicit Constructions and Algorithmic Aspects The problem of finding a good explicit construction is often very difficult. Even the simple proof of Erdős that there are red/blue colorings of graphs with 2 k/2 nodes containing no monochromatic clique of size k leads to an open problem that seems very difficult. An Open Problem Can we explicitly construct a graph as described above with n (1 + ɛ) k nodes in time that is polynomial in n? This problem is still wide open, despite considerable efforts from many mathematicians.
36 Thank You! References: 1 Alon, Spencer, The Probabilistic Method. 2 Gowers, et al., The Princeton Companion to Mathematics. 3 Molloy, Reed, Graph Colouring and the Probabilistic Method.
The application of probabilistic method in graph theory. Jiayi Li
The application of probabilistic method in graph theory Jiayi Li Seite 2 Probabilistic method The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul
More informationThe Probabilistic Method
The Probabilistic Method Ted, Cole, Reilly, Manny Combinatorics Math 372 November 23, 2015 Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, 2015 1 / 18 Overview 1 History created by
More informationRandomness and Computation
Randomness and Computation or, Randomized Algorithms Mary Cryan School of Informatics University of Edinburgh RC (2018/19) Lecture 11 slide 1 The Probabilistic Method The Probabilistic Method is a nonconstructive
More informationLecture 5: January 30
CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 5: January 30 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationIndependence numbers of locally sparse graphs and a Ramsey type problem
Independence numbers of locally sparse graphs and a Ramsey type problem Noga Alon Abstract Let G = (V, E) be a graph on n vertices with average degree t 1 in which for every vertex v V the induced subgraph
More informationLecture 1 : Probabilistic Method
IITM-CS6845: Theory Jan 04, 01 Lecturer: N.S.Narayanaswamy Lecture 1 : Probabilistic Method Scribe: R.Krithika The probabilistic method is a technique to deal with combinatorial problems by introducing
More informationProbabilistic Method. Benny Sudakov. Princeton University
Probabilistic Method Benny Sudakov Princeton University Rough outline The basic Probabilistic method can be described as follows: In order to prove the existence of a combinatorial structure with certain
More informationRandom Lifts of Graphs
27th Brazilian Math Colloquium, July 09 Plan of this talk A brief introduction to the probabilistic method. A quick review of expander graphs and their spectrum. Lifts, random lifts and their properties.
More informationNear-domination in graphs
Near-domination in graphs Bruce Reed Researcher, Projet COATI, INRIA and Laboratoire I3S, CNRS France, and Visiting Researcher, IMPA, Brazil Alex Scott Mathematical Institute, University of Oxford, Oxford
More informationProbabilistic Methods in Combinatorics Lecture 6
Probabilistic Methods in Combinatorics Lecture 6 Linyuan Lu University of South Carolina Mathematical Sciences Center at Tsinghua University November 16, 2011 December 30, 2011 Balance graphs H has v vertices
More informationHypergraph Ramsey numbers
Hypergraph Ramsey numbers David Conlon Jacob Fox Benny Sudakov Abstract The Ramsey number r k (s, n is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains a red
More informationCertainty from uncertainty: the probabilistic method
Certainty from uncertainty: the probabilistic method Lily Silverstein UC Davis Nov 9, 2017 Seattle University 0 / 17 Ramsey numbers 1 / 17 Ramsey numbers A graph is a set of vertices {1,..., n} and edges
More informationProbabilistic Proofs of Existence of Rare Events. Noga Alon
Probabilistic Proofs of Existence of Rare Events Noga Alon Department of Mathematics Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv, Tel Aviv 69978 ISRAEL 1. The Local Lemma In a typical
More informationMa/CS 6b Class 15: The Probabilistic Method 2
Ma/CS 6b Class 15: The Probabilistic Method 2 By Adam Sheffer Reminder: Random Variables A random variable is a function from the set of possible events to R. Example. Say that we flip five coins. We can
More informationPaul Erdős and Graph Ramsey Theory
Paul Erdős and Graph Ramsey Theory Benny Sudakov ETH and UCLA Ramsey theorem Ramsey theorem Definition: The Ramsey number r(s, n) is the minimum N such that every red-blue coloring of the edges of a complete
More informationProperly colored Hamilton cycles in edge colored complete graphs
Properly colored Hamilton cycles in edge colored complete graphs N. Alon G. Gutin Dedicated to the memory of Paul Erdős Abstract It is shown that for every ɛ > 0 and n > n 0 (ɛ), any complete graph K on
More informationMath 261A Probabilistic Combinatorics Instructor: Sam Buss Fall 2015 Homework assignments
Math 261A Probabilistic Combinatorics Instructor: Sam Buss Fall 2015 Homework assignments Problems are mostly taken from the text, The Probabilistic Method (3rd edition) by N Alon and JH Spencer Please
More informationCS5314 Randomized Algorithms. Lecture 18: Probabilistic Method (De-randomization, Sample-and-Modify)
CS5314 Randomized Algorithms Lecture 18: Probabilistic Method (De-randomization, Sample-and-Modify) 1 Introduce two topics: De-randomize by conditional expectation provides a deterministic way to construct
More informationA New Variation of Hat Guessing Games
A New Variation of Hat Guessing Games Tengyu Ma 1, Xiaoming Sun 1, and Huacheng Yu 1 Institute for Theoretical Computer Science Tsinghua University, Beijing, China Abstract. Several variations of hat guessing
More informationRamsey theory. Andrés Eduardo Caicedo. Undergraduate Math Seminar, March 22, Department of Mathematics Boise State University
Andrés Eduardo Department of Mathematics Boise State University Undergraduate Math Seminar, March 22, 2012 Thanks to the NSF for partial support through grant DMS-0801189. My work is mostly in set theory,
More informationApplications of the Lopsided Lovász Local Lemma Regarding Hypergraphs
Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided
More informationThe Probabilistic Method
The Probabilistic Method Janabel Xia and Tejas Gopalakrishna MIT PRIMES Reading Group, mentors Gwen McKinley and Jake Wellens December 7th, 2018 Janabel Xia and Tejas Gopalakrishna Probabilistic Method
More informationLower Bounds on Classical Ramsey Numbers
Lower Bounds on Classical Ramsey Numbers constructions, connectivity, Hamilton cycles Xiaodong Xu 1, Zehui Shao 2, Stanisław Radziszowski 3 1 Guangxi Academy of Sciences Nanning, Guangxi, China 2 School
More informationTHE METHOD OF CONDITIONAL PROBABILITIES: DERANDOMIZING THE PROBABILISTIC METHOD
THE METHOD OF CONDITIONAL PROBABILITIES: DERANDOMIZING THE PROBABILISTIC METHOD JAMES ZHOU Abstract. We describe the probabilistic method as a nonconstructive way of proving the existence of combinatorial
More informationOn the Maximum Number of Hamiltonian Paths in Tournaments
On the Maximum Number of Hamiltonian Paths in Tournaments Ilan Adler Noga Alon Sheldon M. Ross August 2000 Abstract By using the probabilistic method, we show that the maximum number of directed Hamiltonian
More informationAll Ramsey numbers for brooms in graphs
All Ramsey numbers for brooms in graphs Pei Yu Department of Mathematics Tongji University Shanghai, China yupeizjy@16.com Yusheng Li Department of Mathematics Tongji University Shanghai, China li yusheng@tongji.edu.cn
More informationThe Lovász Local Lemma : A constructive proof
The Lovász Local Lemma : A constructive proof Andrew Li 19 May 2016 Abstract The Lovász Local Lemma is a tool used to non-constructively prove existence of combinatorial objects meeting a certain conditions.
More informationGraph Packing - Conjectures and Results
Graph Packing p.1/23 Graph Packing - Conjectures and Results Hemanshu Kaul kaul@math.iit.edu www.math.iit.edu/ kaul. Illinois Institute of Technology Graph Packing p.2/23 Introduction Let G 1 = (V 1,E
More informationInduced subgraphs of prescribed size
Induced subgraphs of prescribed size Noga Alon Michael Krivelevich Benny Sudakov Abstract A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(g denote the maximum
More informationOn the chromatic number and independence number of hypergraph products
On the chromatic number and independence number of hypergraph products Dhruv Mubayi Vojtĕch Rödl January 10, 2004 Abstract The hypergraph product G H has vertex set V (G) V (H), and edge set {e f : e E(G),
More informationChromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz
Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz Jacob Fox Choongbum Lee Benny Sudakov Abstract For a graph G, let χ(g) denote its chromatic number and σ(g) denote
More informationNew Results on Graph Packing
Graph Packing p.1/18 New Results on Graph Packing Hemanshu Kaul hkaul@math.uiuc.edu www.math.uiuc.edu/ hkaul/. University of Illinois at Urbana-Champaign Graph Packing p.2/18 Introduction Let G 1 = (V
More informationPacking nearly optimal Ramsey R(3, t) graphs
Packing nearly optimal Ramsey R(3, t) graphs He Guo Joint work with Lutz Warnke Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue edge-coloring of complete n-vertex
More informationOlympiad Combinatorics. Pranav A. Sriram
Olympiad Combinatorics Pranav A. Sriram August 2014 Chapter 9: The Probabilistic Method 1 Copyright notices All USAMO and USA Team Selection Test problems in this chapter are copyrighted by the Mathematical
More informationInduced subgraphs with many repeated degrees
Induced subgraphs with many repeated degrees Yair Caro Raphael Yuster arxiv:1811.071v1 [math.co] 17 Nov 018 Abstract Erdős, Fajtlowicz and Staton asked for the least integer f(k such that every graph with
More informationDominating a family of graphs with small connected subgraphs
Dominating a family of graphs with small connected subgraphs Yair Caro Raphael Yuster Abstract Let F = {G 1,..., G t } be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive
More informationZero-sum square matrices
Zero-sum square matrices Paul Balister Yair Caro Cecil Rousseau Raphael Yuster Abstract Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the
More informationAsymptotically optimal induced universal graphs
Asymptotically optimal induced universal graphs Noga Alon Abstract We prove that the minimum number of vertices of a graph that contains every graph on vertices as an induced subgraph is (1+o(1))2 ( 1)/2.
More information(n 1)/2. Proof. Choose a random tournament. The chance that it fails to have S k is bounded above by this probability.
: Preliminaries Asymptotic notations f = O(g) means f cg. f = o(g) means f/g 0. f = Ω(g) means f cg. f = Θ(g) means c 1 g f c g. f g means f/g 1. 1 p e p for all nonnegative p, close for small p. The Stirling
More informationOn the Maximum Number of Hamiltonian Paths in Tournaments
On the Maximum Number of Hamiltonian Paths in Tournaments Ilan Adler, 1 Noga Alon,, * Sheldon M. Ross 3 1 Department of Industrial Engineering and Operations Research, University of California, Berkeley,
More informationF 2k 1 = F 2n. for all positive integers n.
Question 1 (Fibonacci Identity, 15 points). Recall that the Fibonacci numbers are defined by F 1 = F 2 = 1 and F n+2 = F n+1 + F n for all n 0. Prove that for all positive integers n. n F 2k 1 = F 2n We
More informationComputing the Independence Polynomial: from the Tree Threshold Down to the Roots
1 / 16 Computing the Independence Polynomial: from the Tree Threshold Down to the Roots Nick Harvey 1 Piyush Srivastava 2 Jan Vondrák 3 1 UBC 2 Tata Institute 3 Stanford SODA 2018 The Lovász Local Lemma
More informationarxiv: v1 [math.co] 2 Dec 2013
What is Ramsey-equivalent to a clique? Jacob Fox Andrey Grinshpun Anita Liebenau Yury Person Tibor Szabó arxiv:1312.0299v1 [math.co] 2 Dec 2013 November 4, 2018 Abstract A graph G is Ramsey for H if every
More informationPacking nearly optimal Ramsey R(3, t) graphs
Packing nearly optimal Ramsey R(3, t) graphs He Guo Georgia Institute of Technology Joint work with Lutz Warnke Context of this talk Ramsey number R(s, t) R(s, t) := minimum n N such that every red/blue
More informationA = A U. U [n] P(A U ). n 1. 2 k(n k). k. k=1
Lecture I jacques@ucsd.edu Notation: Throughout, P denotes probability and E denotes expectation. Denote (X) (r) = X(X 1)... (X r + 1) and let G n,p denote the Erdős-Rényi model of random graphs. 10 Random
More informationREU Problems. Instructor: László Babai Last updated: Monday, July 11 (Problems added)
REU 2011 - Problems Instructor: László Babai e-mail: laci@cs.uchicago.edu Last updated: Monday, July 11 (Problems 48-53 added) 1. (Balancing numbers) Suppose we have 13 real numbers with the following
More information1 Notation. 2 Sergey Norin OPEN PROBLEMS
OPEN PROBLEMS 1 Notation Throughout, v(g) and e(g) mean the number of vertices and edges of a graph G, and ω(g) and χ(g) denote the maximum cardinality of a clique of G and the chromatic number of G. 2
More information< k 2n. 2 1 (n 2). + (1 p) s) N (n < 1
List of Problems jacques@ucsd.edu Those question with a star next to them are considered slightly more challenging. Problems 9, 11, and 19 from the book The probabilistic method, by Alon and Spencer. Question
More informationRamsey-type problem for an almost monochromatic K 4
Ramsey-type problem for an almost monochromatic K 4 Jacob Fox Benny Sudakov Abstract In this short note we prove that there is a constant c such that every k-edge-coloring of the complete graph K n with
More informationComplementary Ramsey numbers, graph factorizations and Ramsey graphs
Complementary Ramsey numbers, graph factorizations and Ramsey graphs Akihiro Munemasa Tohoku University joint work with Masashi Shinohara May 30, 2017, Tohoku University 1st Tohoku-Bandung Bilateral Workshop:
More informationChromatic number, clique subdivisions, and the conjectures of
Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz Jacob Fox Choongbum Lee Benny Sudakov MIT UCLA UCLA Hajós conjecture Hajós conjecture Conjecture: (Hajós 1961) If
More informationAsymptotically optimal induced universal graphs
Asymptotically optimal induced universal graphs Noga Alon Abstract We prove that the minimum number of vertices of a graph that contains every graph on vertices as an induced subgraph is (1 + o(1))2 (
More informationStability of the path-path Ramsey number
Worcester Polytechnic Institute Digital WPI Computer Science Faculty Publications Department of Computer Science 9-12-2008 Stability of the path-path Ramsey number András Gyárfás Computer and Automation
More informationARRANGEABILITY AND CLIQUE SUBDIVISIONS. Department of Mathematics and Computer Science Emory University Atlanta, GA and
ARRANGEABILITY AND CLIQUE SUBDIVISIONS Vojtěch Rödl* Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 and Robin Thomas** School of Mathematics Georgia Institute of Technology
More informationPigeonhole Principle and Ramsey Theory
Pigeonhole Principle and Ramsey Theory The Pigeonhole Principle (PP) has often been termed as one of the most fundamental principles in combinatorics. The familiar statement is that if we have n pigeonholes
More informationNotes 6 : First and second moment methods
Notes 6 : First and second moment methods Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Roc, Sections 2.1-2.3]. Recall: THM 6.1 (Markov s inequality) Let X be a non-negative
More informationDiscrete mathematics , Fall Instructor: prof. János Pach
Discrete mathematics 016-017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations, 1.7.
More informationLecture 8: February 8
CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 8: February 8 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationLinear Algebra Methods in Combinatorics
Linear Algebra Methods in Combinatorics Arjun Khandelwal, Joshua Xiong Mentor: Chiheon Kim MIT-PRIMES Reading Group May 17, 2015 Arjun Khandelwal, Joshua Xiong May 17, 2015 1 / 18 Eventown and Oddtown
More informationLarge Cliques and Stable Sets in Undirected Graphs
Large Cliques and Stable Sets in Undirected Graphs Maria Chudnovsky Columbia University, New York NY 10027 May 4, 2014 Abstract The cochromatic number of a graph G is the minimum number of stable sets
More informationA Study on Ramsey Numbers and its Bounds
Annals of Pure and Applied Mathematics Vol. 8, No. 2, 2014, 227-236 ISSN: 2279-087X (P), 2279-0888(online) Published on 17 December 2014 www.researchmathsci.org Annals of A Study on Ramsey Numbers and
More informationThe Erdős-Hajnal hypergraph Ramsey problem
The Erdős-Hajnal hypergraph Ramsey problem Dhruv Mubayi Andrew Suk February 28, 2016 Abstract Given integers 2 t k +1 n, let g k (t, n) be the minimum N such that every red/blue coloring of the k-subsets
More information1 Primals and Duals: Zero Sum Games
CS 124 Section #11 Zero Sum Games; NP Completeness 4/15/17 1 Primals and Duals: Zero Sum Games We can represent various situations of conflict in life in terms of matrix games. For example, the game shown
More informationMa/CS 6b Class 12: Graphs and Matrices
Ma/CS 6b Class 2: Graphs and Matrices 3 3 v 5 v 4 v By Adam Sheffer Non-simple Graphs In this class we allow graphs to be nonsimple. We allow parallel edges, but not loops. Incidence Matrix Consider a
More informationAcyclic subgraphs with high chromatic number
Acyclic subgraphs with high chromatic number Safwat Nassar Raphael Yuster Abstract For an oriented graph G, let f(g) denote the maximum chromatic number of an acyclic subgraph of G. Let f(n) be the smallest
More informationThe Lopsided Lovász Local Lemma
Department of Mathematics Nebraska Wesleyan University With Linyuan Lu and László Székely, University of South Carolina Note on Probability Spaces For this talk, every a probability space Ω is assumed
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh 27 June 2008 1 Warm-up 1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets
More informationOut-colourings of Digraphs
Out-colourings of Digraphs N. Alon J. Bang-Jensen S. Bessy July 13, 2017 Abstract We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring.
More informationConstructions in Ramsey theory
Constructions in Ramsey theory Dhruv Mubayi Andrew Suk Abstract We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform
More informationCS6999 Probabilistic Methods in Integer Programming Randomized Rounding Andrew D. Smith April 2003
CS6999 Probabilistic Methods in Integer Programming Randomized Rounding April 2003 Overview 2 Background Randomized Rounding Handling Feasibility Derandomization Advanced Techniques Integer Programming
More informationDiscrete Mathematics
Discrete Mathematics Workshop Organized by: ACM Unit, ISI Tutorial-1 Date: 05.07.2017 (Q1) Given seven points in a triangle of unit area, prove that three of them form a triangle of area not exceeding
More informationGeorgia Tech High School Math Competition
Georgia Tech High School Math Competition Multiple Choice Test February 28, 2015 Each correct answer is worth one point; there is no deduction for incorrect answers. Make sure to enter your ID number on
More informationUniquely Hamiltonian Graphs
Uniquely Hamiltonian Graphs Benedikt Klocker Algorithms and Complexity Group Institute of Computer Graphics and Algorithms TU Wien Retreat Talk Uniquely Hamiltonian Graphs Benedikt Klocker 2 Basic Definitions
More informationATLANTA LECTURE SERIES In Combinatorics and Graph Theory (XIX)
ATLANTA LECTURE SERIES In Combinatorics and Graph Theory (XIX) April 22-23, 2017 GEORGIA STATE UNIVERSITY Department of Mathematics and Statistics Sponsored by National Security Agency and National Science
More informationTHE PROBABILISTIC METHOD IN COMBINATORICS
THE PROBABILISTIC METHOD IN COMBINATORICS Lectures by Niranjan Balachandran. Contents 1 The Probabilistic Method: Some First Examples 5 1.1 Lower Bounds on the Ramsey Number R(n, n)............... 5 1.2
More informationUSING GRAPHS AND GAMES TO GENERATE CAP SET BOUNDS
USING GRAPHS AND GAMES TO GENERATE CAP SET BOUNDS JOSH ABBOTT AND TREVOR MCGUIRE Abstract. Let F 3 be the field with 3 elements and consider the k- dimensional affine space, F k 3, over F 3. A line of
More information(a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict.)
1 Enumeration 11 Basic counting principles 1 June 2008, Question 1: (a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict) n/2 ( ) n (b) Find a closed form
More informationINTERMINGLED ASCENDING WAVE M-SETS. Aaron Robertson Department of Mathematics, Colgate University, Hamilton, New York
INTERMINGLED ASCENDING WAVE M-SETS Aaron Robertson Department of Mathematics, Colgate University, Hamilton, New York arobertson@colgate.edu and Caitlin Cremin, Will Daniel, and Quer Xiang 1 Abstract Given
More informationSome Applications of pq-groups in Graph Theory
Some Applications of pq-groups in Graph Theory Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 g-exoo@indstate.edu January 25, 2002 Abstract
More informationMonochromatic and Rainbow Colorings
Chapter 11 Monochromatic and Rainbow Colorings There are instances in which we will be interested in edge colorings of graphs that do not require adjacent edges to be assigned distinct colors Of course,
More informationCS 125 Section #12 (More) Probability and Randomized Algorithms 11/24/14. For random numbers X which only take on nonnegative integer values, E(X) =
CS 125 Section #12 (More) Probability and Randomized Algorithms 11/24/14 1 Probability First, recall a couple useful facts from last time about probability: Linearity of expectation: E(aX + by ) = ae(x)
More informationGraph Theory. Thomas Bloom. February 6, 2015
Graph Theory Thomas Bloom February 6, 2015 1 Lecture 1 Introduction A graph (for the purposes of these lectures) is a finite set of vertices, some of which are connected by a single edge. Most importantly,
More informationRamsey Unsaturated and Saturated Graphs
Ramsey Unsaturated and Saturated Graphs P Balister J Lehel RH Schelp March 20, 2005 Abstract A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number,
More informationThe number of edge colorings with no monochromatic cliques
The number of edge colorings with no monochromatic cliques Noga Alon József Balogh Peter Keevash Benny Sudaov Abstract Let F n, r, ) denote the maximum possible number of distinct edge-colorings of a simple
More informationZERO-SUM ANALOGUES OF VAN DER WAERDEN S THEOREM ON ARITHMETIC PROGRESSIONS
ZERO-SUM ANALOGUES OF VAN DER WAERDEN S THEOREM ON ARITHMETIC PROGRESSIONS Aaron Robertson Department of Mathematics, Colgate University, Hamilton, New York arobertson@colgate.edu Abstract Let r and k
More informationTheorem (Special Case of Ramsey s Theorem) R(k, l) is finite. Furthermore, it satisfies,
Math 16A Notes, Wee 6 Scribe: Jesse Benavides Disclaimer: These notes are not nearly as polished (and quite possibly not nearly as correct) as a published paper. Please use them at your own ris. 1. Ramsey
More informationThe expansion of random regular graphs
The expansion of random regular graphs David Ellis Introduction Our aim is now to show that for any d 3, almost all d-regular graphs on {1, 2,..., n} have edge-expansion ratio at least c d d (if nd is
More informationInduced subgraphs of Ramsey graphs with many distinct degrees
Induced subgraphs of Ramsey graphs with many distinct degrees Boris Bukh Benny Sudakov Abstract An induced subgraph is called homogeneous if it is either a clique or an independent set. Let hom(g) denote
More informationSize and degree anti-ramsey numbers
Size and degree anti-ramsey numbers Noga Alon Abstract A copy of a graph H in an edge colored graph G is called rainbow if all edges of H have distinct colors. The size anti-ramsey number of H, denoted
More informationLecture 5: Probabilistic tools and Applications II
T-79.7003: Graphs and Networks Fall 2013 Lecture 5: Probabilistic tools and Applications II Lecturer: Charalampos E. Tsourakakis Oct. 11, 2013 5.1 Overview In the first part of today s lecture we will
More informationInduced Ramsey-type theorems
Induced Ramsey-type theorems Jacob Fox Benny Sudakov Abstract We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and
More informationBIRTHDAY PROBLEM, MONOCHROMATIC SUBGRAPHS & THE SECOND MOMENT PHENOMENON / 23
BIRTHDAY PROBLEM, MONOCHROMATIC SUBGRAPHS & THE SECOND MOMENT PHENOMENON Somabha Mukherjee 1 University of Pennsylvania March 30, 2018 Joint work with Bhaswar B. Bhattacharya 2 and Sumit Mukherjee 3 1
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh June 2009 1 Linear independence These problems both appeared in a course of Benny Sudakov at Princeton, but the links to Olympiad problems are due to Yufei
More informationNotes on the Probabilistic Method
Notes on the Probabilistic Method First Edition: May 6, 009 Anders Sune Pedersen Dept. of Mathematics & Computer Science University of Southern Denmark Campusvej 55, 530 Odense M, Denmark asp@imada.sdu.dk
More informationApplications of Eigenvalues in Extremal Graph Theory
Applications of Eigenvalues in Extremal Graph Theory Olivia Simpson March 14, 201 Abstract In a 2007 paper, Vladimir Nikiforov extends the results of an earlier spectral condition on triangles in graphs.
More informationThis section is an introduction to the basic themes of the course.
Chapter 1 Matrices and Graphs 1.1 The Adjacency Matrix This section is an introduction to the basic themes of the course. Definition 1.1.1. A simple undirected graph G = (V, E) consists of a non-empty
More informationMATH 682 Notes Combinatorics and Graph Theory II
MATH 68 Notes Combinatorics and Graph Theory II 1 Ramsey Theory 1.1 Classical Ramsey numbers Furthermore, there is a beautiful recurrence to give bounds on Ramsey numbers, but we will start with a simple
More informationInduced subgraphs of graphs with large chromatic number. IX. Rainbow paths
Induced subgraphs of graphs with large chromatic number. IX. Rainbow paths Alex Scott Oxford University, Oxford, UK Paul Seymour 1 Princeton University, Princeton, NJ 08544, USA January 20, 2017; revised
More informationPainting Squares in 2 1 Shades
Painting Squares in 1 Shades Daniel W. Cranston Landon Rabern May 1, 014 Abstract Cranston and Kim conjectured that if G is a connected graph with maximum degree and G is not a Moore Graph, then χ l (G
More informationLower bounds for Ramsey numbers for complete bipartite and 3-uniform tripartite subgraphs
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 7, no. 6, pp. 67 688 (03 DOI: 0.755/jgaa.003 Lower bounds for Ramsey numbers for complete bipartite and 3-uniform tripartite subgraphs
More information