Erdős Problem #866 - Discussion thread

Let $k\geq 3$ and $g_k(N)$ be minimal such that if $A\subseteq \{1,\ldots,2N\}$ has $\lvert A\rvert \geq N+g_k(N)$ then there exist integers $b_1,\ldots,b_k$ such that all $\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$).

Estimate $g_k(N)$.

#866: [CES75][Er92c,p.41]

number theory | additive combinatorics

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A problem of Choi, Erdős, and Szemerédi. It is clear that, for the set of odd numbers in $\{1,\ldots,2N\}$, no such $b_i$ exist, whence $g_k(N)\geq 0$ always. Choi, Erdős, and Szemerédi proved that $g_3(N)=2$ and $g_4(N) \ll 1$. van Doorn has shown that $g_4(N)\leq 2338$.

Choi, Erdős, and Szemerédi also proved that\[g_5(N)\asymp \log N\]and\[g_6(N)\asymp N^{1/2}.\]In general they proved that\[g_k(N) \ll_k N^{1-2^{-k}}\]and for every $\epsilon>0$ if $k$ is sufficiently large then\[g_k(N) > N^{1-\epsilon}.\]As an example, taking $A$ to be the set of all odd integers and the powers of $2$ shows that $g_5(N)\gg \log N$.

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Related OEIS sequences: Possible

Additional thanks to: Wouter van Doorn

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #866, https://www.erdosproblems.com/866, accessed 2025-11-15

1 comment on this problem

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I got interested in trying to find an explicit upper bound for $g_4(N)$. It should still be considered a work in progress, but what I have written so far can be found here, and gives an upper bound of $g_4(N) \le 2032$. If I find any improvements (or mistakes that change my currently claimed bound), or if I decide to put it on the arXiv, I'll be sure to update this comment.

Woett — 17:10 on 30 Aug 2025

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