  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 the three problems of Burr and Erd\H{o}s on Ramsey complete
sequences, for which Erd\H{o}s 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 Erd\H{o}s 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 Erd\H{o}s and Graham on sets of integers avoiding a given subset
sum; and, answering a question reiterated by several authors, a homogeneous
strengthening of a seminal result of Szemer\'edi and Vu on long arithmetic
progressions in subset sums.
