The h-critical number and sumsets of nonbases of maximum size

Abstract

Let G be a finite abelian group. Then the h-critical number is the smallest positive integer m such that each subset of G of size m is a basis of order h. In the thesis, we provide the h-critical number for every h and every G using several proofs. We also look at the possible sizes of the sumset hA, whenever a subset A of G is of size m-1, or a nonbasis of order h of maximum size. We give a complete answer for h=2 and h=3.