1 - Introduction to hypergraphs

We begin with an introduction to hypergraphs, which gives a taste of different representations of hypergraphs, linear hypergraphs, and Turán-type problems, including existence of Turán densities and classification of zero Turán densities. Thereafter we delve deeper into some of the classical theorems of hypergraph theory, including various theorems on intersecting families such as Sperner’s Theorem, the LYM Inequality, the Erdős-Ko-Rado Theorem, Hilton-Milner Theorem, Deza-Frankl Theorem, Erdős-Rado Theorem and Frankl-Wilson Theorem. The tools include linear algebraic methods, polynomial methods, the delta-system method, compressions and shadows. The general Turán problem is considered in the framework of analytic methods and Lagrangians, and we consider some specific case studies where the exact answers are known.