We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of $350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdős on the minimum number of colors needed to color the positive integers less than n so that n cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdős and Graham on sets of integers avoiding a given subset sum; and, answering a question of Tran, Vu and Wood, a homogeneous strengthening of a seminal result of Szemerédi and Vu on long arithmetic progressions in subset sums.

Joint work with Jacob Fox and Huy Tuan Pham.