Optimal bounds for an Erdős problem on matching integers to distinct multiples

Abstract: Let $f(m)$ be the largest integer such that for every set $A = {a_1 < \cdots < a_m}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in I$. Solving a problem of Erdős, we determine $f(m)$ exactly, and show $$ f(m)=\min\bigl(m,\lceil 2\sqrt{m}\,\rceil\bigr) $$ for all $m$. The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written. [This paper solves Problem #650 on Bloom's website "Erdős problems".]

• The paper presents f(m)=min(m, ⌈2√m⌉) as the exact bound for disjoint matching pairs in intervals of length 2aₘ, resolving a long-standing Erdős conjecture.
• It employs combinatorial constructions and advanced graph-theoretic techniques, including a generalization of Hall’s theorem, to tightly bound the number of divisible pairs.
• The study integrates AI-assisted proof strategies with formal verification via Lean, demonstrating the potential of automation in complex mathematical proofs.

Optimal Bounds for Matchings of Integers to Distinct Multiples: Resolution of an Erdős Problem

Introduction and Problem Statement

The paper addresses a significant problem in combinatorial number theory originally posed by Erdős: for a set $A$ of $m$ positive integers, and for any open interval $I$ of length $2a_m$ (where $a_m = \max A$), what is the maximal integer $f(m)$ such that there always exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$, $b \in I$, and $m$0? The main result provides an exact determination of $m$1 for all $m$2:

$m$3

for all $m$4. This resolves conjectures and bounds posed by Erdős and others in the literature, which until now were separated by a factor of two.

Historical Context and Prior Work

The lower and upper bounds for $m$5 were initiated by Erdős and Surányi (who proved $m$6) and by Erdős and Selfridge (who established $m$7, giving the general upper bound $m$8). Erdős later questioned whether the lower bound could be improved and how tightly $m$9 could be estimated within these bounds, listed as Problem #650 on Bloom’s compilation.

Main Theorem and Proof Architecture

The proof of the optimal bound $I$0 is split into upper and lower bound arguments:

• Upper Bound ($I$1): For arbitrary $I$2, the construction is based on partitioning $I$3 as $I$4 and building sets $I$5 (size $I$6) and intervals $I$7 where no more than $I$8 distinct divisible pairs can occur. The construction uses properties of least common multiples, residue classes, and the Chinese Remainder Theorem, leading to the tight bound.
• Lower Bound ($I$9): This follows via a graph-theoretic approach, associating to each instance a bipartite graph with partite sets $2a_m$0 and $2a_m$1 (the integers in the interval), where edges represent divisibility. A generalization of Hall's theorem (the Kőnig–Ore formula) gives a matching of nearly the desired size by bounding the neighborhood sizes appropriately using injections and refined combinatorial estimates.

A key technical result is that for all subsets $2a_m$2, the size of the neighborhood $2a_m$3 in the bipartite graph satisfies $2a_m$4, enabling the lower bound construction for the matching.

Technical Ingredients

The argument refines and improves upon earlier combinatorial and number-theoretic techniques by:

• Implementing careful interval splitting and the construction of sets $2a_m$5 with explicit arithmetic structure,
• Exploiting prime moduli to control divisibility conflicts,
• Using generalizations of Hall's marriage theorem for lower-bounding maximum matchings,
• Incorporating injections based on difference patterns to bound the image size within the product of neighborhood sizes,
• Generalizing the interval length to values $2a_m$6, with the main result holding throughout this range.

The technical proofs are rigorous and demonstrate that previous upper and lower bounds are indeed tight.

AI-Assisted Proof Development and Formal Verification

A notable aspect is the AI-assisted workflow in the proof's discovery and verification:

• An initial proof strategy was generated by ChatGPT (GPT-5.4 Pro), which correctly identified the combinatorial structure, although it left critical details open.
• The Lean formalization, conducted by the Aristotle system, both filled gaps and checked details, with some aspects of the proof being adapted to ensure correctness after AI-suggested approaches failed to give injections in special cases.

This demonstrates the emerging paradigm of AI-proposed strategies followed by formal machine verification, with the Lean proof publicly available and the process outlined in detail.

Implications and Future Directions

This result conclusively resolves a long-standing Erdős problem about the structure of multiples in intervals relative to sparse sets, precisely quantifying the matching number. The interplay between number theory and extremal combinatorics is deepened, and the result reinforces the precision attainable in multiplicative combinatorics. Practically, this clarifies constraints on possible matchings in multiplicative systems and may inform related investigations in additive number theory, graph theory, and Ramsey-type problems.

On the methodological side, the AI-assisted and formally certified workflow for proof discovery, refinement, and verification is showcased as a viable approach for intricate combinatorial problems, suggesting further developments in AI-theorem proving pipelines for challenging mathematical conjectures.

Conclusion

The paper achieves an exact determination of $2a_m$7, the maximal number of matching pairs in intervals of length $2a_m$8 where each pair consists of an element of $2a_m$9 and its distinct multiple. By demonstrating that $a_m = \max A$0 for all $a_m = \max A$1, longstanding uncertainties are resolved. The proof combines advanced combinatorial constructions, rigorous graph-theoretic arguments, and formal computer-assisted verification, with broader implications for both mathematical theory and future AI–human workflows in pure mathematics.