Probabilistic Methods in Combinatorics

This lecture introduces the Erdős-Ko-Rado problem, which deals with intersecting families of sets. The instructor explains how to determine the maximum size of such families and presents examples to illustrate the concept. The lecture also covers the scenario of picking fellowships from a pool of applicants, focusing on selecting a group where no individual is better than the entire group. Various claims and probabilities related to pairwise comparisons and 'bad events' are discussed in the context of selecting a valid winner. The instructor emphasizes the importance of avoiding 'impossible scenarios' in the selection process.