Extremal combinatorics explores how large or small a finite collection of combinatorial objects can be while satisfying certain conditions. For example, how many edges can a graph without triangles contain? Or, how many distinct positive integers can be in a sequence without creating any increasing or decreasing subsequence of length k? This broad area naturally intersects many other fields of study, such as number theory, algebra, geometry, and probability. In this talk, we will survey some of the classical results in extremal combinatorics with a focus on problems from graph theory.
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