Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs

Three classes of finite structures are related by extremal properties: complete d-partite d-uniform hypergraphs, d-dimensional affine cubes of integers, and families of 2d sets forming a d-dimensional Boolean algebra. We review extremal results for each of these classes and derive new ones for Boolean algebras and hypergraphs, several obtained by employing relationships between the three classes. Related partition or coloring problems are also studied for Boolean algebras. Density results are given for Boolean algebras of sets all of whose atoms are the same size.