BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Huy Pham (Stanford University) DTSTART:20220524T170000Z DTEND:20220524T172500Z DTSTAMP:20260423T011320Z UID:CANT2022/7 DESCRIPTION:Title: Homogeneous structures in subset sums and applications\nby Huy Pham ( Stanford University) as part of Combinatorial and additive number theory ( CANT 2022)\n\n\nAbstract\nIn recent joint works with David Conlon and Jaco b Fox\, we develop novel techniques which allow us to prove a diverse rang e of results relating to subset sums. In the one-dimensional case\, our te chniques imply the existence of long homogeneous arithmetic progressions i n the set of subset sums under a variety of assumptions. This allows us to resolve a number of longstanding open problems\, including: solutions to the three problems of Burr and Erdos on Ramsey complete sequences\, for wh ich Erdos later offered a combined total of 350\; analogous results for th e new notion of density complete sequences\; the solution to a conjecture of Alon and Erdos on the minimum number of colors needed to color the posi tive integers less than n so that n cannot be written as a monochromatic s um\; the exact determination of an extremal function introduced by Erdos a nd Graham on sets of integers avoiding a given subset sum\; and\, answerin g a question reiterated by several authors\, a homogeneous strengthening o f a result of Szemeredi and Vu on long arithmetic progressions in subset s ums. In follow-up work in the multi-dimensional case\, we show the existen ce of large homogeneous generalized arithmetic progressions in the set of subset sums of sufficiently large subsets of [n]\, yielding a strengthenin g of a seminal result of Szemeredi and Vu. As an application\, we make pro gress on the Erdos--Straus non-averaging sets problem\, showing that every subset A of [n] of size at least n^{\\sqrt{2} - 1 + o(1)} contains an ele ment which is the average of two or more other elements of A. This gives t he first polynomial improvement on a result of Erdos and Sarkozy from 1990 .\n LOCATION:https://researchseminars.org/talk/CANT2022/7/ END:VEVENT END:VCALENDAR