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[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: Oftentimes, in applying the probabilistic method, we'll need to understand whether a certain random structure typically has a property that we're looking for. In this video, we'll look at one such property and understand how to analyze whether a certain random graph has the property of containing a triangle with high probability. The objects that we're going to be looking at today is the Erdos-Rényi random graph, commonly referred to as G(n,p). And this is the graph consisting of placing n vertices. And for each pair of vertices, we put down an edge between them with probability p independently for all pairs of vertices. In other words, we flip a biased coin that comes up heads with probability p independently for each possible edge and put down the edge randomly as such. This is a random graph. It's a random structure. It's a random object. And we would like to understand various properties of this ra...

[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: In this video, we'll look at three basic yet important inequalities in probability-- Markov's inequality, Chebyshev's inequality, and the Chernoff bound. Markov's inequality says that if we are given X, a real valued non-negative random variable, then for every positive number lambda we have the following inequality-- the probability that X is bigger-- is at least lambda, this probability is no more than the expectation of X divided by lambda. One way to interpret this inequality is that if X is a non-negative random variable with small expectation, then it is unlikely for X to be very high. This is a very important and useful inequality. And let us prove it. The proof is quite short. We can start with the expectation of X. And rewrite this in the following way. It is at least the expectation of X times the indicator function corresponding to when X is at least lambda. So this means it is 1 if X is at least lam...

[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: In this video, let us look at a basic yet important concept in probability known as linearity of expectations and use it to deduce some interesting consequences in combinatorics via the probabilistic method. Linearity of expectations says that, if you are given random variables, x1 to xn, and constants, C1 through Cn, then when we take a linear combination of these random variables, as such, C1 times x1-- so imagine these are real valued random variables and these are real constants, for instance-- then the sum of these C1 times x1 plus C2 times x2 and so on plus Cn times xn-- so this sum has expectation-- the following, which can be computed by distributing this expectation symbol across to the individual variables. So this is a basic and important property. And it's worth noting that a similar statement written for products is often not true. So it is not usually the case that the expectation of a product of two random v...

[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: In this video, we'll look at an application of the probabilistic method to graph theory. An independent set in a graph is a subset of vertices with no two adjacent. For example, if the graph is this cycle on four vertices, an example of an independent set would be two vertices like this. And these two vertices do not-- these two vertices are not adjacent to each other, whereas had I chosen two vertices on the same edge, that would not be an independent set. So an important question in graph theory is given a graph, what can you say about the size of its independent sets? The following theorem, due to Caro-Wei, says that every graph G contains a large independent set in the following sense. It contains an independent set of size at least the following quantity, summing over all v among vertices G, 1 over the degree of v plus 1. So let us prove this theorem first, and then we'll see an application and some ways to interp...

[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: In this video, we'll look at an application of the probabilistic method to extremal set theory. Extremal set theory is an area of combinatorics concerned with questions such as, what is the largest collection of sets that satisfy certain desired properties? And the question that we'll look at in this video is the following. What is the largest antichain? And more precisely, what is the largest collection of subsets of some ground set numbers 1 through n-- so this is an n element set. So I want as many sets as we can such that none of these sets is a subset of another set. So this is what we mean by an antichain. And for a given value of n, what is the largest number of sets you can have? What is the largest l that we can have as a function of n? To give an example, when n equals to 3, you can have the sets 1, 2; 1, 3; and 2, 3; and you see that none of these sets is a superset of another. More generally, when you take ...

[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: Let us look at an application of the probabilistic method to extremal set theory. The extremal set theory concerns the study of families of sets that have certain desirable properties, and asking questions such as what is the largest set family that you can have with such families-- with such properties? And in this video, we'll focus on the following property of being an intersecting family. This means a collection of sets, A1 through Al, such that these sets pairwise intersect in a nonempty intersection. So we want sets A1 through Al, such that whenever you take any two of these sets and take their intersection, it is never the empty set. So here are two basic questions that one can ask. The first one, which will turn out to be an easy warm up, is what is the largest intersecting family of subsets of 1 through n. So here, there are, in total, 2 to the n subsets of 1 through n. But if I want to only keep a collection of s...

[SQUEAKING] [RUSTLING] [CLICKING] YUFEI ZHAO: In this video, let us look at an application of the probabilistic method to graph theory. We'll prove what is known as the crossing number inequality. Now what is the crossing number of a graph? If I give you a graph, sometimes it's possible to draw the graph on a plane without crossings. For example, the complete graph on four vertices can be drawn on the plane without having any pair of edges crossed. Such graph is called a planar graph. But sometimes, it's not possible to draw such a graph on the plane. And the classic example of such a graph is K5, the complete graph on five vertices. And then in that case, maybe you want to know what's the minimum number of edge crossings you can have in any possible drawing of this graph? So the crossing number of a graph is defined to be the minimum number of edge crossings of a drawing of G of this graph G on the plane using continuous curves as edges. It is a classic f...

YUFEI ZHAO: We've been spending the past few lectures discussing Szemeredi's Regularity Lemma. And one of the first applications that we discussed of the Regularity Lemma is the triangle removal Lemma. So today, I want to revisit this topic and show you a strengthening of the Removal Lemma for which new regularity techniques are needed. But first, recall the graph removal Lemma. In the graph removal Lemma, we have that for every graph H and epsilon bigger than zero, there exists some delta such that if an N vertex graph has fewer than delta and to the number of vertices of H, many copies of H, then it can be made H-free by removing fewer than epsilon N squared edges. Even in the case when H is a triangle, when this is called a triangle removal Lemma, even in that case, basically the regularity method is more or less the only way that we currently know how to prove this theorem. So we saw this a few lectures ago. What I would like to discuss today is a variant of t...