Probabilistic Methods in Combinatorics - Fall 2024

Explore graduate-level probabilistic methods in combinatorics through this comprehensive MIT course that introduces fundamental techniques for proving the existence of combinatorial objects using random constructions. Master the core methodology of showing that random constructions work with positive probability, covering essential topics including large bipartite subgraphs, Ramsey number lower bounds, extremal set theory through Sperner's theorem and intersecting families, linearity of expectations, independent sets and Turán's theorem, crossing number inequalities, and concentration inequalities including Markov, Chebyshev, Chernoff, and bounded differences inequalities. Delve into advanced applications such as determining thresholds for random graphs containing triangles and proving the existence of graphs with both high girth and high chromatic number. Gain proficiency in these powerful probabilistic techniques that form the foundation of modern combinatorics and theoretical computer science, with emphasis on both theoretical methodology and practical combinatorial applications taught by instructor Yufei Zhao.