## View abstract

### Invited talk

Tuesday, July 20, 14:45 ~ 15:45 UTC-3

## Subset sums, completeness and colorings

### Jacob Fox

#### Stanford University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak640e76c8499b7c084e5ce71df1faae34').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy640e76c8499b7c084e5ce71df1faae34 = 'j&#97;c&#111;bf&#111;x' + '&#64;'; addy640e76c8499b7c084e5ce71df1faae34 = addy640e76c8499b7c084e5ce71df1faae34 + 'st&#97;nf&#111;rd' + '&#46;' + '&#101;d&#117;'; var addy_text640e76c8499b7c084e5ce71df1faae34 = 'j&#97;c&#111;bf&#111;x' + '&#64;' + 'st&#97;nf&#111;rd' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak640e76c8499b7c084e5ce71df1faae34').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy640e76c8499b7c084e5ce71df1faae34 + '\'>'+addy_text640e76c8499b7c084e5ce71df1faae34+'<\/a>';

In this talk, we discuss novel techniques which allow us to prove a diverse range of results on representing integers as subset sums, including solutions to several long-standing open problems in the area. These include: solutions to the three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of \$350 for their solution; 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 reiterated by several authors, a homogeneous strengthening of a seminal result of Szemerédi and Vu on long arithmetic progressions in subset sums.

Joint work with David Conlon (Caltech, USA) and Huy Tuan Pham (Stanford University, USA).