In 2001 Croot resolved an old conjecture of Erdős and Graham, proving that in any finite colouring of the positive integers there is a (non-trivial) monochromatic solution to 1/n1+…+1/nk = 1 with all ni distinct. A natural generalisation, also conjectured by Erdos and Graham, is that in fact any set of positive density contains such a solution. We will discuss the proof of this conjecture, which extends Croot’s method, and uses Fourier analysis coupled with elementary number theoretic and combinatorial arguments.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
