🎲Extremal Combinatorics Unit 5 – Probabilistic Method & Random Graphs

Key Concepts and Definitions

• Probabilistic method a powerful tool in combinatorics that proves the existence of certain objects by showing that a randomly chosen object has the desired properties with positive probability
• Random graph a graph generated by a random process, often used to model complex networks and analyze their properties
• Erdős–Rényi model denoted as $G(n, p)$, generates a random graph with $n$ vertices where each edge is included independently with probability $p$
• Threshold function a function $p(n)$ such that a property holds with high probability for $p(n) \gg p_0(n)$ and fails with high probability for $p(n) \ll p_0(n)$
  • Helps determine the emergence of certain graph properties as the edge probability increases
• Connectivity the property of a graph being connected, meaning there is a path between any two vertices
• Giant component a connected component that contains a constant fraction of the vertices in a random graph
• Clique a complete subgraph where every pair of vertices is connected by an edge
• Independent set a set of vertices in a graph where no two vertices are adjacent

Probabilistic Method Fundamentals

• Basic idea prove the existence of an object with certain properties by showing that a randomly constructed object has those properties with positive probability
• Nonconstructive proofs the probabilistic method often proves the existence of objects without explicitly constructing them
• Expectation the average value of a random variable, used to analyze the properties of random structures
  • Linearity of expectation a powerful tool that allows the expectation of a sum of random variables to be computed as the sum of their individual expectations, even if the variables are dependent
• Markov's inequality a fundamental inequality that bounds the probability of a non-negative random variable exceeding a certain value
  • For a non-negative random variable $X$ and $a > 0$, $\Pr[X \geq a] \leq \frac{\mathbb{E}[X]}{a}$
• Chebyshev's inequality a stronger inequality that bounds the probability of a random variable deviating from its mean by a certain amount
  • For a random variable $X$ with mean $\mu$ and variance $\sigma^2$, and $k > 0$, $\Pr[|X - \mu| \geq k\sigma] \leq \frac{1}{k^2}$
• Chernoff bounds a family of concentration inequalities that provide tight bounds on the probability of a sum of independent random variables deviating from its expected value
• Lovász Local Lemma a powerful tool for proving the existence of objects that satisfy a set of local constraints, even when the constraints are not independent

Random Graph Models

• Erdős–Rényi model $G(n, p)$ generates a random graph with $n$ vertices where each edge is included independently with probability $p$
  • $G(n, m)$ variant generates a random graph with $n$ vertices and $m$ edges, chosen uniformly at random from all possible graphs with these parameters
• Preferential attachment model generates a growing random graph where new vertices are more likely to connect to existing vertices with high degrees (rich-get-richer effect)
• Small-world model generates a random graph with high clustering and short average path lengths, capturing properties observed in many real-world networks
• Random regular graphs graphs where each vertex has the same degree, chosen uniformly at random from all possible regular graphs with the given degree sequence
• Inhomogeneous random graphs allow for different vertices to have different expected degrees, capturing the heterogeneity observed in many real-world networks
• Hypergraphs generalize the concept of graphs by allowing edges to connect more than two vertices, enabling the modeling of more complex relationships
• Random geometric graphs generated by placing vertices randomly in a metric space (Euclidean space) and connecting pairs of vertices that are within a certain distance threshold

Proof Techniques and Strategies

• First moment method uses the expectation of a random variable to prove the existence of an object with a desired property
  • If $\mathbb{E}[X] > 0$ for a non-negative integer-valued random variable $X$, then $\Pr[X > 0] > 0$, implying the existence of an object counted by $X$
• Second moment method uses the variance of a random variable to prove the existence of an object with a desired property
  • Provides stronger concentration results than the first moment method by considering the second moment (variance) of the random variable
• Alteration method starts with a random object and modifies it step by step to obtain an object with the desired properties
  • Useful when direct analysis of the final object is difficult
• Polynomial method represents combinatorial objects as polynomials and uses algebraic techniques to prove the existence of objects with specific properties
• Janson's inequality bounds the probability of the union of a set of events in a random graph, taking into account their dependencies
• Talagrand's inequality a concentration inequality for functions of independent random variables that satisfy certain boundedness and Lipschitz conditions
• Szemerédi's Regularity Lemma a powerful tool in graph theory that partitions a large graph into a bounded number of nearly random bipartite subgraphs, enabling the application of probabilistic arguments

Applications in Extremal Combinatorics

• Ramsey theory studies the emergence of ordered substructures in large structures, often using the probabilistic method to prove the existence of Ramsey numbers
  • Ramsey number $R(k, l)$ is the smallest integer $n$ such that any red-blue coloring of the edges of the complete graph $K_n$ contains either a red $K_k$ or a blue $K_l$
• Turán's theorem determines the maximum number of edges in a graph that does not contain a specific subgraph (Turán graph)
  • Probabilistic proofs provide alternative derivations and generalizations of Turán's theorem
• Szemerédi's theorem states that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions
  • Probabilistic proofs using the regularity lemma and the hypergraph removal lemma
• Extremal graph theory studies the maximum or minimum values of graph parameters under certain constraints, often using probabilistic arguments to establish bounds
• Discrepancy theory investigates the deviation of a coloring of a set system from a uniform coloring, using probabilistic methods to prove the existence of colorings with low discrepancy
• Additive combinatorics explores the behavior of subsets of additive structures (integers, finite fields) under arithmetic operations, employing probabilistic techniques to establish structural results
• Coding theory uses probabilistic arguments to prove the existence of codes with good properties (error-correcting codes, locally decodable codes)

Important Theorems and Results

• Erdős-Kac theorem describes the asymptotic distribution of the number of prime factors of a randomly chosen integer, demonstrating the central limit theorem in number theory
• Erdős-Ko-Rado theorem determines the maximum size of a family of subsets of a finite set, where any two subsets have at least one element in common
  • Probabilistic proofs provide short and elegant derivations of the theorem
• Erdős-Stone theorem a generalization of Turán's theorem that determines the asymptotic behavior of the Turán number for a given graph $H$
• Ajtai-Komlós-Szemerédi theorem bounds the independence number of a triangle-free graph with a given average degree, using the probabilistic method
• Kahn-Kalai conjecture (now a theorem) states that the expectation of the product of two non-negative random variables is at most the product of their expectations, with applications in combinatorics and probability theory
• Friedgut's sharp threshold theorem shows that many graph properties exhibit a sharp threshold behavior, transitioning from holding with low probability to holding with high probability over a small range of edge probabilities
• Szemerédi's regularity lemma partitions a large graph into a bounded number of nearly random bipartite subgraphs, enabling the application of probabilistic arguments in graph theory

Problem-Solving Examples

• Proving the existence of Ramsey numbers using the probabilistic method
  • Show that a random red-blue coloring of the edges of a large complete graph contains either a large red or blue complete subgraph with high probability
• Establishing lower bounds on the Turán number for a given graph $H$ using random graphs
  • Construct a random graph with a given edge density and show that it does not contain $H$ as a subgraph with high probability
• Proving the existence of graphs with specific properties using the probabilistic method
  • Construct a random graph with a given edge probability and show that it satisfies the desired properties with positive probability
• Applying the Lovász Local Lemma to prove the existence of colorings or other combinatorial objects that satisfy a set of local constraints
  • Construct a random coloring or object and show that the probability of violating any constraint is small enough to apply the Local Lemma
• Using the alteration method to prove the existence of graphs with specific substructures
  • Start with a random graph and modify it step by step to create the desired substructures while maintaining the required properties
• Applying Janson's inequality to bound the probability of the existence of certain subgraphs in a random graph
  • Express the event of interest as the union of a set of dependent events and apply Janson's inequality to obtain a concentration result
• Using Talagrand's inequality to establish concentration results for functions of independent random variables in a combinatorial setting
  • Verify that the function of interest satisfies the boundedness and Lipschitz conditions required by Talagrand's inequality

Connections to Other Areas

• Number theory probabilistic methods are used to study the distribution of prime numbers, arithmetic progressions, and other number-theoretic objects
  • Erdős-Kac theorem, Szemerédi's theorem, and the Green-Tao theorem on arithmetic progressions in the primes
• Computer science probabilistic algorithms and randomized data structures rely on ideas from the probabilistic method
  • Randomized quicksort, skip lists, and locality-sensitive hashing
• Coding theory probabilistic arguments are used to prove the existence of error-correcting codes with good properties and to analyze the performance of decoding algorithms
  • Random linear codes, low-density parity-check codes, and locally decodable codes
• Machine learning probabilistic models and techniques are central to many machine learning algorithms, particularly in unsupervised learning and generative modeling
  • Bayesian networks, Markov random fields, and variational autoencoders
• Theoretical computer science probabilistic methods are used to analyze the performance of algorithms, prove lower bounds, and establish the complexity of computational problems
  • Randomized algorithms, probabilistically checkable proofs, and hardness of approximation
• Statistical physics ideas from the probabilistic method are used to study the behavior of large complex systems, such as spin glasses and percolation models
  • Gibbs measures, phase transitions, and the cavity method
• Network science random graph models and probabilistic techniques are used to analyze the structure and dynamics of complex networks, such as social networks and biological networks
  • Preferential attachment models, small-world networks, and community detection algorithms