MAT 307 : Combinatorics Lecture 11 : The probabilistic method
نویسنده
چکیده
1 Probability basics A probability space is a pair (Ω,Pr) where Pr is a normalized measure on Ω, i.e. Pr(Ω) = 1. In combinatorics, it’s mostly sufficient to work with finite probability spaces, so we can avoid a lot of the technicalities of measure theory. We can assume that Ω is a finite set and each elementary event ω ∈ Ω has a certain probability Pr[ω] ∈ [0, 1]; ω∈Ω Pr[ω] = 1. Any subset A ⊆ Ω is an event, of probability Pr[A] = ω∈A Pr[ω]. Observe that a union of events corresponds to OR and an intersection of events corresponds to AND. A random variable is any function X : Ω → R. Two important notions here will be expectation and independence. Definition 1. The expectation of a random variable X is E[X] = ∑ ω∈Ω X(ω) Pr[ω] = ∑
منابع مشابه
MAT 307 : Combinatorics Lecture 9 - 10 : Extremal combinatorics
1 Bipartite forbidden subgraphs We have seen the Erdős-Stone theorem which says that given a forbidden subgraph H, the extremal number of edges is ex(n,H) = 2(1−1/(χ(H)−1)+o(1))n. Here, o(1) means a term tending to zero as n → ∞. This basically resolves the question for forbidden subgraphs H of chromatic number at least 3, since then the answer is roughly cn2 for some constant c > 0. However, f...
متن کاملRandom Graphs and the Probabilistic Method
These notes accompany a lecture given in the summer of 2006 at the Center for Combinatorics at Nankai University. They are provided as a reference (and especially bibliography) for students new to the probabilistic method. The goal in these notes is twofold. First we wish to demonstrate the basics of the probabilistic method, covering both the expectation and second moment applications. Second,...
متن کامل2016 Lecture 7 : The Probabilistic Method
We now move on to a very interesting and powerful technique in combinatorics, called the probabilistic method. There is an excellent (really excellent!) textbook written by Noga Alon and Joel Spencer [AS16]. In fact, most materials that we will cover are from [AS16]. The method works as follows. Our goal is to prove the existence of certain object or structure with desired properties. We start ...
متن کاملThe Probabilistic Method in Combinatorics
2 The Linearity of Expectation and small tweakings 11 2.1 Revisiting the Ramsey Number R(n, n) . . . . . . . . . . . . . . . . . . . 11 2.2 List Chromatic Number and minimum degree . . . . . . . . . . . . . . . 11 2.3 The Daykin-Erdős conjecture . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 An Example from Combinatorial Geometry . . . . . . . . . . . . . . . . . 14 2.5 Graphs with High...
متن کاملMAT 307 : Combinatorics Lecture 14 : Topological methods
1 The Borsuk-Ulam theorem We have seen how combinatorics borrows from probability theory. Another area which has been very beneficial to combinatorics, perhaps even more surprisingly, is topology. We have already seen Brouwer's fixed point theorem and its combinatorial proof. Theorem 1 (Brouwer). For any continuous function f : B n → B n , there is a point x ∈ B n such that f (x) = x. A more po...
متن کامل