We develop novel techniques which allow us to prove a diverse range of results around subset sums, 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 colours needed to colour 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 David Conlon and Jacob Fox.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
