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Erdős problem 131

Let F(N)F(N) be the maximal size of A⊆{1,…,N}A\subseteq\{1,\ldots,N\} such that no a∈Aa\in A divides the sum of any nonempty subset of A∖{a}A\setminus\{a\}. Estimate F(N)F(N). The lower bound F(N)≫N1/5F(N)\gg N^{1/5} is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave F(N)≤N1/4+o(1)F(N)\leq N^{1/4+o(1)}. The claimed new result is the matching upper bound F(N)≤N1/5+o(1)F(N)\leq N^{1/5+o(1)}, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence F(N)=N1/5+o(1)F(N)=N^{1/5+o(1)}.

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