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

Ricky Cipollini

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

组合数学 2026-06-28 v1 数论

作者: Ricky Cipollini

摘要

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 Erd\H{o}s 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.

引用

@article{arxiv.2606.29361,
  title  = {A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem},
  author = {Ricky Cipollini},
  journal= {arXiv preprint arXiv:2606.29361},
  year   = {2026}
}