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This lecture is the third part of a short course on "The Probabilistic Method" delivered by Robert Morris from IMPA (Instituto de Matemática Pura e Aplicada) during the 2nd Brazilian School of Combinatorics (EBC). Explore fundamental concepts of the probabilistic method in combinatorial mathematics through this 75-minute session. The course is part of a larger event featuring 11 plenary talks, 2 mini-courses, and poster presentations, designed to foster collaboration among researchers and students from Brazil and abroad. The 2nd EBC will be held at IMPA in Rio de Janeiro from March 10-14, 2025, continuing the success of the first edition which gathered 87 participants from 8 Brazilian states and 10 countries.
