A beginner's guide to edge and cover ideals

Monomial ideals, although intrinsically interesting, play an important role in studying the connections between commutative algebra and combinatorics. Broadly speaking, problems in combinatorics are encoded into monomial ideals, which then allow us to use techniques and methods in commutative algebra to solve the original question. Stanley's proof of the Upper Bound Conjecture [180] for simplicial spheres is seen as one of the early highlights of exploiting this connection between two fields. To bridge these two areas of mathematics, Stanley used square-free monomial ideals. © 2013 Springer-Verlag Berlin Heidelberg.