DOCSLIB.ORG

The Probabilistic Method in Graph Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Probabilistic Method in Graph Theory Introduction Applications in Discrete Mathematics Some examples and results The Probabilistic Method In Graph Theory Ehssan Khanmohammadi Department of Mathematics The Pennsylvania State University February 25, 2010 Introduction Applications in Discrete Mathematics Some examples and results What do we mean by the probabilistic method? Why use this method? The Probabilistic Method and Paul Erd}os 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 Erd}oswho applied it mainly to problems in combinatorics and number theory from 1947 onwards. Introduction Applications in Discrete Mathematics Some examples and results 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 Erd}osin 1960s and 1970s, there was at least one talk which concluded with Erd}os informing the speaker that almost every graph was a counterexample to his/her conjecture! Introduction Applications in Discrete Mathematics Some examples and results 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 1010, nodes. Explicit constructions necessarily introduce some structuredness to the class of graphs built, which thus restricts the graphs considered. 3: Covers almost Every Graph Erd}osdid not say some graph is a counterexample to your conjecture, but rather almost every graph is a counterexample to your conjecture. Introduction Applications in Discrete Mathematics Some examples and results What do we mean by the probabilistic method? Why use this method? 2: Powerful and Easy to Use Erd}oswould routinely perform the necessary calculations to disprove a conjecture in his head during a fifteen-minute talk. Introduction Applications in Discrete Mathematics Some examples and results What do we mean by the probabilistic method? Why use this method? 2: Powerful and Easy to Use Erd}oswould routinely perform the necessary calculations to disprove a conjecture in his head during a fifteen-minute talk. 3: Covers almost Every Graph Erd}osdid not say some graph is a counterexample to your conjecture, but rather almost every graph is a counterexample to your conjecture. Introduction Applications in Discrete Mathematics Some examples and results 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? Surprisingly often it is much easier to prove this than it is to give an example of a structure that works. Introduction Applications in Discrete Mathematics Some examples and results 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. Introduction Applications in Discrete Mathematics Some examples and results 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. 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. Introduction Applications in Discrete Mathematics Some examples and results 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. Introduction Applications in Discrete Mathematics Some examples and results 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. 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 Xσ be the indicator of \σ defines a Hamiltonian path". −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Introduction Applications in Discrete Mathematics Some examples and results Szele's result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. 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 Xσ be the indicator of \σ defines a Hamiltonian path". −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Introduction Applications in Discrete Mathematics Some examples and results 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. Let Xσ be the indicator of \σ defines a Hamiltonian path". −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Introduction Applications in Discrete Mathematics Some examples and results 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. −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Introduction Applications in Discrete Mathematics Some examples and results 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 Xσ be the indicator of \σ defines a Hamiltonian path". P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Introduction Applications in Discrete Mathematics Some examples and results 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 Xσ be the indicator of \σ defines a Hamiltonian path". −(n−1) P(Xσ = 1) = 2 . Conclusion: There is a tournament for which X is equal to at least E(X ). Introduction Applications in Discrete Mathematics Some examples and results 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 Xσ be the indicator of \σ defines a Hamiltonian path". −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Introduction Applications in Discrete Mathematics Some examples and results 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 Xσ be the indicator of \σ defines a Hamiltonian path". −(n−1) P(Xσ = 1) = 2 . P P −(n−1) X = σ Xσ, thus E(X ) = σ E(Xσ) = n!2 . Conclusion: There is a tournament for which X is equal to at least E(X ). 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. Introduction Applications in Discrete Mathematics Some examples and results Remark 1 A player who wins all games would naturally be the tournament's winner.
Load more

Recommended publications
  • Information Theory
    Information Theory Gurkirat,Harsha,Parth,Puneet,Rahul,Tushant November 8, 2015 1 Introduction[1] Information theory is a branch of applied mathematics, electrical engineering, and computer science involving the quantification of information . Firstly, it provides a general way to deal with any information , be it language, music, binary codes or pictures by introducing a measure of information , en- tropy as the average number of bits(binary for our convenience) needed to store one symbol in a message. This abstraction of information through entropy allows us to deal with all kinds of information mathematically and apply the same principles without worrying about it's type or form! It is remarkable to note that the concept of entropy was introduced by Ludwig Boltzmann in 1896 as a measure of disorder in a thermodynamic system , but serendipitously found it's way to become a basic concept in Information theory introduced by Shannon in 1948 . Information theory was developed as a way to find fundamental limits on the transfer , storage and compression of data and has found it's way into many applications including Data Compression and Channel Coding. 2 Basic definitions 2.1 Intuition behind "surprise" or "information"[2] Suppose someone tells you that the sun will come up tomorrow.This probably isn't surprising. You already knew the sun will come up tomorrow, so they didn't't really give you much information. Now suppose someone tells you that aliens have just landed on earth and have picked you to be the spokesperson for the human race. This is probably pretty surprising, and they've given you a lot of information.The rarer something is, the more you've learned if you discover that it happened.
    [Show full text]
  • Probabilistic Methods in Combinatorics
    Lecture notes (MIT 18.226, Fall 2020) Probabilistic Methods in Combinatorics Yufei Zhao Massachusetts Institute of Technology [email protected] http://yufeizhao.com/pm/ Contents 1 Introduction7 1.1 Lower bounds to Ramsey numbers......................7 1.1.1 Erdős’ original proof..........................8 1.1.2 Alteration method...........................9 1.1.3 Lovász local lemma........................... 10 1.2 Set systems................................... 11 1.2.1 Sperner’s theorem............................ 11 1.2.2 Bollobás two families theorem..................... 12 1.2.3 Erdős–Ko–Rado theorem on intersecting families........... 13 1.3 2-colorable hypergraphs............................ 13 1.4 List chromatic number of Kn;n ......................... 15 2 Linearity of expectations 17 2.1 Hamiltonian paths in tournaments....................... 17 2.2 Sum-free set................................... 18 2.3 Turán’s theorem and independent sets.................... 18 2.4 Crossing number inequality.......................... 20 2.4.1 Application to incidence geometry................... 22 2.5 Dense packing of spheres in high dimensions................. 23 2.6 Unbalancing lights............................... 26 3 Alterations 28 3.1 Ramsey numbers................................ 28 3.2 Dominating set in graphs............................ 28 3.3 Heilbronn triangle problem........................... 29 3.4 Markov’s inequality............................... 31 3.5 High girth and high chromatic number.................... 31 3.6 Greedy random coloring............................ 32 2 4 Second moment method 34 4.1 Threshold functions for small subgraphs in random graphs......... 36 4.2 Existence of thresholds............................. 42 4.3 Clique number of a random graph....................... 47 4.4 Hardy–Ramanujan theorem on the number of prime divisors........ 49 4.5 Distinct sums.................................. 52 4.6 Weierstrass approximation theorem...................... 54 5 Chernoff bound 56 5.1 Discrepancy..................................
    [Show full text]
  • 14 the Probabilistic Method
    14 The Probabilistic Method Probabilistic Graph Theory Theorem 14.1 (Szele) For every positive integer n, there exists a tournament on n vertices with at least n!2¡(n¡1) Hamiltonian paths. Proof: Construct a tournament by randomly orienting each edge of Kn in each direction 1 independently with probability 2 . For any permutation σ of the vertices, let Xσ be the indicator random variable which has value 1 if σ is the vertex sequence of a Hamiltonian path, and 0 otherwise. Let X be the random variable which is the total number of Hamiltonain P paths. Then X = Xσ and we have X ¡(n¡1) E(X) = E(Xσ) = n!2 : σ Thus, there must exist at least one tournament on n vertices which has ¸ n!2¡(n¡1) Hamil- tonain paths as claimed. ¤ 1 Theorem 14.2 Every graph G has a bipartite subgraph H for which jE(H)j ¸ 2 jE(G)j. Proof: Choose a random subset S ⊆ V (G) by independently choosing each vertex to be in S 1 with probability 2 . Let H be the subgraph of G containing all of the vertices, and all edges with exactly one end in S. For every edge e, let Xe be the indicator random variable with value 1 if e 2 E(H) and 0 otherwise. If e = uv, then e will be in H if u 2 S and v 62 S or if 1 u 62 S and v 2 S, so E(Xe) = P(e 2 E(H)) = 2 and we ¯nd X 1 E(jE(H)j) = E(Xe) = 2 jE(G)j: e2E(G) 1 Thus, there must exist at least one bipartite subgraph H with jE(H)j ¸ 2 jE(G)j.
    [Show full text]
  • 18.218 Probabilistic Method in Combinatorics, Topic 3: Alterations
    Lecturer: Prof. Yufei Zhao Notes by: Andrew Lin 3 Alterations Recall the naive probabilistic method: we found some lower bounds for Ramsey numbers in Section 1.1, primarily for the diagonal numbers. We did this with a basic method: color randomly, so that we color each edge red with probability p and blue with probability 1 − p. Then the probability that we don’t see any red s-cliques or blue t-cliques (with a union bound) is at most n (s ) n (t ) p 2 + (1 − p) 2 ; s t and if this is less than 1 for some p, then there exists some graph on n vertices for which there is no red Ks and blue Kt . So we union bounded the bad events there. Well, the alteration method does a little more than that - here’s a proof that mirrors that of Proposition 1.6. We again color randomly, but the idea now is to delete a vertex in every bad clique (red Ks and blue Kt ). How many edges have we deleted? We can estimate by using linearity of expectation: Theorem 3.1 For all p 2 (0; 1); n 2 N, n (s ) n (t ) R(s; t) > n − p 2 − (1 − p) 2 : s t This right hand side begins by taking the starting number of vertices and then we deleting one vertex for each clique. We’re going to explore this idea of “fixing the blemishes” a little more. 3.1 Dominating sets Definition 3.2 Given a graph G, a dominating set U is a set of vertices such that every vertex not in U has a neighbor in U.
    [Show full text]
  • Modern Discrete Probability an Essential Toolkit
    Modern Discrete Probability An Essential Toolkit Sebastien´ Roch November 20, 2014 Modern Discrete Probability: An Essential Toolkit Manuscript version of November 20, 2014 Copyright c 2014 Sebastien´ Roch Contents Preface vi I Introduction 1 1 Models 2 1.1 Preliminaries ............................ 2 1.1.1 Review of graph theory ................... 2 1.1.2 Review of Markov chain theory .............. 2 1.2 Percolation ............................. 2 1.3 Random graphs ........................... 2 1.4 Markov random fields ........................ 2 1.5 Random walks on networks ..................... 3 1.6 Interacting particles on finite graphs ................ 3 1.7 Other random structures ...................... 3 II Basic tools: moments, martingales, couplings 4 2 Moments and tails 5 2.1 First moment method ........................ 6 2.1.1 The probabilistic method .................. 6 2.1.2 Markov’s inequality .................... 8 2.1.3 . Random permutations: longest increasing subsequence . 9 2.1.4 . Constraint satisfaction: bound on random k-SAT threshold 11 2.1.5 . Percolation on Zd: existence of a phase transition .... 12 2.2 Second moment method ...................... 16 2.2.1 Chebyshev’s and Paley-Zygmund inequalities ....... 17 2.2.2 . Erdos-R¨ enyi´ graphs: small subgraph containment .... 21 2.2.3 . Erdos-R¨ enyi´ graphs: connectivity threshold ....... 25 i 2.2.4 . Percolation on trees: branching number, weighted sec- ond moment method, and application to the critical value . 28 2.3 Chernoff-Cramer´ method ...................... 32 2.3.1 Tail bounds via the moment-generating function ..... 33 2.3.2 . Randomized algorithms: Johnson-Lindenstrauss, "-nets, and application to compressed sensing .......... 37 2.3.3 . Information theory: Shannon’s theorem ......... 44 2.3.4 .
    [Show full text]
  • GIRTH and CHROMATIC NUMBER of GRAPHS Contents 1
    GIRTH AND CHROMATIC NUMBER OF GRAPHS AARON HALPER Abstract. This paper will look at the relationship between high girth and high chromatic number in both its finite and transfinite incarnations. On the one hand, we will demonstrate that it is possible to construct graphs with high oddgirth and high chromatic number in all cases. We will then look at a theorem which tells us why, at least in the transfinite case, it is impossible to generalize this to include even cycles. Finally, we will use the probabilistic method to show why it is possible to construct graphs of any given finite girth and finite chromatic number. Contents 1. Introduction 1 2. Oddgirth and Chromatic Number 2 3. Graphs with Uncountable Chromatic Number 4 4. The Probabilistic Method 7 Acknowledgements 9 References 9 1. Introduction We begin with a few basic definitions: Definition 1.1. A graph G is defined as a set of vertices V = fv1; v2; :::; vng and edges E that connect the vertices to each other. We signify that two vertices v; w are connected by writing fv; wg. Note that even though G is here defined as having only a countable number of vertices, we will later allow G to have an uncountable number of vertices. Definition 1.2. On any graph G, we can impose a coloring wherein we color each of the vertices in such a way that if two vertices are connected to each other, then they cannot have the same color. Definition 1.3. The Chromatic Number χ of a graph is the smallest number of colors needed to fully color the graph.
    [Show full text]
  • The Probabilistic Method
    The Probabilistic Method Camilo Espinosa May 16, 2018 Contents 1 Background 2 1.1 Probability Basics . 2 1.2 Useful Approximations . 3 2 The Basic Probabilistic Method 4 2.1 Description of the Method . 4 2.2 Ramsey Numbers . 4 2.3 Hypergraph Colorings . 5 2.4 Hamiltonian Paths . 5 3 The Lovasz Local Lemma 6 3.1 The General Lovasz Local Lemma . 6 3.2 The Symmetric Lovasz Local Lemma . 8 3.3 Cycles in a Directed Graph . 9 4 Further Work 10 5 References 10 1 Abstract In this paper, we will examine a group of techniques that fall under the umbrella of the Probabilistic Method and will solve several problems in order to illustrate how the method is used. We will first introduce the probabilistic and algebraic concepts necessary to understand the method, followed by a description of the workflow the method entails. We will then examine applications of the method to prove statements about Ramsey Numbers, specific colorings of hypergraphs, and Hamiltonian paths in directed graphs. Afterwards, we will delve into advanced applications of the method by stating and proving the Symmetric Lovasz Local Lemma, which we will use to show an interesting result on cycles in directed graphs. We will then briefly discuss some of the newer results that have followed the Lovasz Local Lemma, specifically the Algorithmic Lovasz Local Lemma and a problem that can be solved in polynomial time because of this theorem. 1 Background 1.1 Probability Basics Definition 1.1.1: A Finite Probability Space is defined as a pair (Ω; P) where Ω is a set of elements ! called elementary
    [Show full text]
  • APPLICATIONS of PROBABILITY THEORY to GRAPHS Contents 1. Introduction 1 2. Preliminaries 2 3. the Probabilistic Method 4 4. Rand
    APPLICATIONS OF PROBABILITY THEORY TO GRAPHS JORDAN LAUNE Abstract. In this paper, we summarize the probabilistic method and survey some of its iconic proofs. We then go on to draw the connections between - regular pairs and random graphs, emphasizing their relationship and parallel constructions. We finish with a review of the proof of Szemer´edi'sRegularity Lemma and an example of its application. Contents 1. Introduction 1 2. Preliminaries 2 3. The Probabilistic Method 4 4. Random Graphs and -Regularity 7 5. Szemer´ediRegularity Lemma 10 Acknowledgments 13 References 13 1. Introduction Some results in graph theory could be proven by having a computer check all graphs possible on a specific vertex set. However, this method of brute force is usually completely out of the question: some problems would require a computer to run for longer than the current age of the universe. Other constructive solutions that include algorithms to construct a graph are usually too complicated to find. Thus, probability theory, which gives us an indirect way to prove properties of graphs, is a very powerful tool in graph theory. Some graph theorists use probability theory as a separate tool to apply to graphs, as in the probabilistic method. On the other hand, graph theorists are sometimes interested in the interesection of graph theory and probabilistic models themselves, such as random graphs. While the probabilistic method was interested in using probability merely as a tool for graphs, those studying the random graph are instead interested in its properties for its own sake. As it turns out, the random graph gives us an idea of how a \general" graph behaves.
    [Show full text]
  • Probabilistic Methods in Combinatorics
    Probabilistic Methods in Combinatorics Joshua Erde Department of Mathematics, Universit¨atHamburg. Contents 1 Preliminaries 5 1.1 Probability Theory . .5 1.2 Useful Estimates . .8 2 The Probabilistic Method 10 2.1 Ramsey Numbers . 10 2.2 Set Systems . 11 3 The First Moment Method 15 3.1 Hamiltonian Paths in a Tournament . 16 3.2 Tur´an'sTheorem . 17 3.3 Crossing Number of Graphs . 18 4 Alterations 20 4.1 Ramsey Numbers Again . 20 4.2 Graphs of High Girth and High Chromatic Numbers . 21 5 Dependent random choice 24 5.1 Tur´anNumbers of Bipartite Graphs . 25 1 5.2 The Ramsey Number of the Cube . 26 5.3 Improvements . 27 6 The Second Moment Method 29 6.1 Variance and Chebyshev's Inequality . 29 6.2 Threshold Functions . 30 6.3 Balanced Subgraphs . 33 7 The Hamiltonicity Threshold 35 7.1 The Connectivity Threshold . 35 7.2 Pos´a'sRotation-Extension Technique . 39 7.3 Hamiltonicty Threshold . 42 8 Strong Concentration 44 8.1 Motivation . 44 8.2 The Chernoff Bound . 44 8.3 Combinatorial Discrepancy . 48 8.4 A Lower Bound for the Binomial Distribution . 49 9 The Lov´asLocal Lemma 52 9.1 The Local Lemma . 52 9.2 Ramsey Bounds for the last time . 55 9.3 Directed Cycles . 56 9.4 The Linear Aboricity of Graphs . 57 10 Martingales and Strong Concentration 62 10.1 The Azuma-Hoeffding Inequality . 62 2 10.2 The Chromatic Number of a Dense Random Graph . 66 10.3 The Chromatic Number of Sparse Random Graphs . 70 11 Talagrand's Inequality 73 11.1 Longest Increasing Subsequence .
    [Show full text]
  • Introduction to the Probabilistic Method
    Introduction to the probabilistic method F. Havet Projet Mascotte, CNRS/INRIA/UNSA, INRIA Sophia-Antipolis, 2004 route des Lucioles BP 93, 06902 Sophia-Antipolis Cedex, France [email protected] The probabilistic method is an efficient technique to prove the existence of combinatorial objects having some specific properties. It is based on the probability theory but it can be used to prove theorems which have nothing to do with probability. The probabilistic method is now widely used and considered as a basic knowledge. It is now exposed in many introductory books to graph theory like [8]. In these notes, we give the most common tools and techniques of this method. The interested reader who would like to know more complicated techniques is referred to the books of Alon and Spencer [5] and Molloy and Reed [21]. We assume that the reader is familiar with basic graph theory. Several books (see e.g. [8, 11, 29]) give nice introduction to graph theory. 1 Basic probabilistic concept A (finite) probability space is a couple (W;Pr) where W is a finite set called sample set, and Pr is a function, called probability function, from W into [0;1] such that ∑w2W Pr(w) = 1. We will 1 often consider a uniform distribution for which Pr(w) = jWj for all w 2 W. The set Gn of the labelled graphs on n vertices may be seen as the sample set of a probability space (Gn;Pr). The result of the selection of an element G of this sample set according to the probability function Pr is called a random graph.
    [Show full text]
  • 18.218 Probabilistic Method in Combinatorics, Lecture Notes
    The Probabilistic Method in Combinatorics Lecturer: Yufei Zhao Notes by: Andrew Lin Spring 2019 This is an edited transcript of the lectures of MIT’s Spring 2019 class 18.218: The Probabilistic Method in Combinatorics, taught by Professor Yufei Zhao. Each section focuses on a different technique, along with examples of applications. Additional course material, including problem sets, can be found on the course website. The main reference for the material is the excellent textbook N. Alon and J. H. Spencer, The probabilistic method, Wiley, 4ed. Most of the course will follow the textbook, though some parts will differ. Special thanks to Abhijit Mudigonda, Mihir Singhal, Andrew Gu, and others for their help in proofreading. Contents 1 Introduction to the probabilistic method 4 1.1 The Ramsey numbers ........................................... 4 1.2 Alterations ................................................. 5 1.3 Lovász Local Lemma ........................................... 6 1.4 Set systems ................................................ 7 1.5 Hypergraph colorings ........................................... 9 2 Linearity of expectation 11 2.1 Setup and basic examples ......................................... 11 2.2 Sum-free sets ............................................... 11 2.3 Cliques ................................................... 12 2.4 Independent sets ............................................. 13 2.5 Crossing numbers ............................................. 14 2.6 Application to incidence geometry ...................................
    [Show full text]
  • Expected Uses of Probability
    Expected Uses of Probability Evan Chen August 11, 2014 Email: [email protected]. This is mostly about expected value, both in its own right and in the context of the probabilistic method. §1 Definitions and Notation Nothing tricky here, just setting up notation. I'll try to not be overly formal. A random variable is just a quantity that we take to vary randomly. For example, the outcome of a standard six-sided dice roll, say D6, is a random variable. We can now discuss the probability of certain events, which we'll denote P(•). For instance, we can write 1 P(D6 = 1) = P(D6 = 2) = ··· = P(D6 = 6) = 6 1 or P(D6 = 0) = 0 and P(D6 ≥ 4) = 2 . We can also discuss the expected value of a random variable X, which is the \average" value. The formal definition is def X E[X] = P(X = x) · x: x But an example for our dice roll D6 makes this clearer: 1 1 1 E[D6] = · 1 + · 2 + ··· + · 6 = 3:5: 6 6 6 In natural language, we just add up all the outcomes weighted by probability they appear. We'll assume the reader has some familiarity with basic graph theory terms; see http:// en.wikipedia.org/wiki/Graph_theory#Definitions otherwise. One term we'll define here that may not be so known { given a graph G, an independent set is a set of vertices for which no two are connected by an edge. §2 Properties of Expected Value §2.1 A Motivating Example It is an unspoken law that any introduction to expected value begins with the following classical example.
    [Show full text]