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Extremal and probabilistic combinatorics. Paper from the 28th Brazilian mathematics colloquium ‒ 28$^{\text o}$ Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 2011. (English)
Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (ISBN 978-85-244-0319-4/pbk). ii, 81~p. (2011).
These lecture notes contain in 81 pages a very brief introduction to extremal and probabilistic combinatorics. The chapters of the book are the following: Preliminaries (numbers and sets, asymptotics, graphs, probability), Ramsey theory (Ramsey’s theorem, Schur’s theorem, finite Ramsey’s theorem, Van der Waerden’s theorem), Extremal graph theory (Turán’s theorem, the Erdős-Stone theorem, counting $H$-free graphs), The random graph ( high girth and chromatic number, Janson’s inequality, the Fortuin, Kasteleyn and Ginibre (FKG) inequality, the giant component), Topological and algebraic methods (the Borsuk-Ulam theorem, the Kneser graph, the Frankl-Wilson inequalities, Borsuk’s conjecture and its solution by Kahn and Kalai), Szemerédi regularity lemma (the Erdős-Turán conjecture, the regularity lemma and its applications, the Erdős-Stone theorem, Roth’s theorem, Ramsey-Turán numbers, graph limits), Dependent random choice (extremal numbers for bipartite graphs, Ramsey-Turán revisited, the Balog-Szemerédi-Gowers theorem). Each chapter includes exercises and recommended further reading. Few results have complete proofs. This book offers a nice landscape on some main chapters of extremal and probabilistic combinatorics.
Reviewer: Ioan Tomescu (Bucureşti)