Preface Notations I Ingredients 1 Number theory 1.1 Divisibility of integers 1.2 Congruences 1.3 The Fundamental Theorem of Number Theory 1.4 Multiplicative number theory 1.5 Additive number theory 2 Combinatorics 2.1 Basic enumeration principles 2.2 Counting lists, sequences, sets, and multisets 2.3 Binomial coefficients and Pascal's Triangle 2.4 Some recurrence relations 2.5 The integer lattice and its layers 3 Group theory 3.1 Finite abelian groups 3.2 Group isomorphisms 3.3 The Fundamental Theorem of Finite Abelian Groups 3.4 Subgroups and cosets 3.5 Subgroups generated by subsets 3.6 Sumsets II Appetizers Spherical designs Caps, centroids, and the game SET How many elements does it take to span a group? In pursuit of perfection The declaration of independence III Sides The function vg (n, h) The function v+(n, h) The function u(n, m, h) The function u^(n, m, h) IV Entrees A Maximum sumset size A.1 Unrestricted sumsets A.1.1 Fixed number of terms A.1.2 Limited number of terms A.1.3 Arbitrary number of terms A.2 Unrestricted signed sumsets A.2.1 Fixed number of terms A.2.2 Limited number of terms A.2.3 Arbitrary numbcr of terms A.3 Restricted sumsets A.3.1 Fixed number of terms A.3.2 Limited number of terms A.3.3 Arbitrary number of terms A.4 Restricted signed sumscts A.4.1 Fixed number of terms A.4.2 Limited number of terms