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arXiv · 2606.29361

A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem

Ricky Cipollini
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Abstract

Let $f_3(N)$ be the least integer such that every set $A\subseteq\{1,\ldots,N\}$ of size at least $f_3(N)$ contains distinct elements $a,b,c\in A$ such that $a+b\in A$, $a+c\in A$, and $b+c\in A$. We prove that $f_3(N)\le 5N/8+O(1)$. Together with the standard construction $[N/8,N/4]\cup[N/2,N]$, this gives $f_3(N)=5N/8+O(1)$, resolving Erdős Problem 865. The proof is self-contained. An earlier conditional version of the reduction has also been formalized in Lean 4/Mathlib with no sorries and no added axioms.

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Ricky Cipollini. 2026-06-28. A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem. https://arxiv.org/abs/2606.29361

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