Subset sums, completeness and colorings

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 long standing open problems. These include: solutions to the three problems of Burr and Erdös on Ramsey complete sequences, for which Erdös later offered a combined total of 350 analogous results for the new notion of density complete sequences; the answer to a question of Alon and Erdös on the minimum number of colors needed to color the positive integers less than so that cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdös and Graham and first studied by Alon on sets of integers avoiding a given subset sum; and, answering a question of Tran, Vu and Wood, a common strengthening of seminal results of Szemerédi-Vu and Sárközy on long arithmetic progressions in subset sums.

Based on joint work with David Conlon and Huy Tuan Pham.