Minors, Embeddability, and Extremal Problems for Hypergraphs

This is an expository chapter based on two talks given during the Special Semester on Discrete and Computational Geometry, organized by János Pach and Emo Welzl, at the École Polytechnique Fédérale in Lausanne, Switzerland, in the fall of 2010. Our first purpose is to describe a circle of ideas regarding topological extremal problems for simplicial complexes (hypergraphs), and the corresponding phase-transition questions for random complexes (as introduced by Linial and Meshulam). In particular, we discuss some notions of minors pertaining to such questions. Our focus is on k-dimensional complexes that are sparse, i.e., such that the number of k-dimensional simplices is linear in the number of (k − 1)-dimensional simplices. Second, we discuss a notion of higher-dimensional expansion for simplicial complexes, due to Gromov, which is very useful in this context (the same notion of expansion has also arisen independently in the work of Linial, Meshulam, and Wallach on random complexes, and in the work of Newman and Rabinovich on higher-dimensional analogues of finite metrics).