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تسجيل مستخدم جديدThe upper logarithmic density of monochromatic subset sums
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We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+sqrt{3})/4$ and this is best possible. This answers a forty-year-old question of ErdH{o}s.
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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 longstanding open problems. These include: solutions to the three probl
ems of Burr and ErdH{o}s on Ramsey complete sequences, for which ErdH{o}s 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 ErdH{o}s 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 ErdH{o}s 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 Szemeredi and Vu on long arithmetic progressions in subset sums.
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