The Probabilistic Method
"Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book."
- MAA Reviews
Maintaining a standard of excellence that establishes The Probabilistic Method as the leading reference on probabilistic methods in combinatorics, the Fourth Edition continues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics.
Emphasizing the methodology and techniques that enable problem-solving, The Probabilistic Method, Fourth Edition begins with a description of tools applied to probabilistic arguments, includingbasic techniques that use expectation and variance as well as the more advanced applications ofmartingales and correlation inequalities. The authors explore where probabilistic techniques havebeen applied successfully and also examine topical coverage such as discrepancy and random graphs,circuit complexity, computational geometry, and derandomization of randomized algorithms.Written by two well-known authorities in the field, the Fourth Edition features:
Additional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques
New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits
Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results
The Probabilistic Method, Fourth Edition is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The Fourth Edition is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory.
Noga Alon, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal Random Structures and Algorithms , Dr. Alon is the recipient of the Polya Prize, The GÃ¶del Prize, The Israel Prize, and the EMET Prize.
Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition , also published by Wiley.
The Probabilistic Method
The Basic Method
What you need is that your brain is open.
1.1 The Probabilistic Method
The probabilistic method is a powerful tool for tackling many problems in discrete mathematics. Roughly speaking, the method works as follows: trying to prove that a structure with certain desired properties exists, one defines an appropriate probability space of structures and then shows that the desired properties hold in these structures with positive probability. The method is best illustrated by examples. Here is a simple one. The Ramsey number is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices by red and blue, either there is a red (i.e., a complete subgraph on k vertices all of whose edges are colored red) or there is a blue . Ramsey 1929 showed that is finite for any two integers k and . Let us obtain a lower bound for the diagonal Ramsey numbers .
If , then . Thus for all .
Consider a random two-coloring of the edges of obtained by coloring each edge independently either red or blue, where each color is equally likely. For any fixed set R of k vertices, let be the event that the induced subgraph of on R is monochromatic (i.e., that either all its edges are red or they are all blue). Clearly, . Since there are possible choices for R , the probability that at least one of the events occurs is at most . Thus, with positive probability, no event occurs and there is a two-coloring of without a monochromatic ; that is, . Note that if and we take , then
and hence for all .
This simple example demonstrates the essence of the probabilistic method. To prove the existence of a good coloring, we do not present one explicitly, but rather show, in a nonconstructive way, that it exists. This example appeared in a paper of P. ErdÃ¶s from 1947. Although Szele had applied the probabilistic method to another combinatorial problem, mentioned in Chapter 2 , already in 1943, ErdÃ¶s was certainly the first to understand the full power of this method and apply it successfully over the years to numerous problems. One can, of course, claim that the probability is not essential in the proof given above. An equally simple proof can be described by counting; we just check that the total number of two-colorings of is larger than the number of those containing a monochromatic .
Moreover, since the vast majority of the probability spaces considered in the study of combinatorial problems are finite, this claim applies to most of the applications of the probabilistic method in discrete mathematics. Theoretically, this is indeed the case. However, in practice the probability is essential. It would be hopeless to replace the applications of many of the tools appearing in this book, including, for example, the second moment method, the LovÃ¡sz Local Lemma and the concentration via martingales by counting arguments, even when these are applied to finite probability spaces.
The probabilistic method has an interesting algorithmic aspect. Consider, for example, the proof of Proposition 1.1.1, which shows that there is an edge two-coloring of without a monochromatic . Can we actually find such a coloring? This question, as asked, may sound ridiculous; the total number of possible colorings is finite, so we can try them all until we find the desired one. However, such a procedure may require steps; an amount of time that is exponential in the size of the problem. Algorithms whose running time is more than polynomial in the size of the problem are usually considered impractical. The class of problems that can be solved in polynomial time, usually denoted by P (see, e.g., Aho, Hopcroft and Ullman 1974), is, in a sense, the class of all solvable probl