ProblemP2988

The limit of the minimum overlap quotient

Problem statement

Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?

Recorded solution requirements

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-36-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1233

R1233Recorded attemptOutcome: Open

Resolve the stated acceptance condition

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Authored summary

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: open strategy

Recorded scope: No scope is recorded.

Originating problem: The limit of the minimum overlap quotient

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Authored record and scope

Authored title: Resolve the stated acceptance condition
Record type: attempt
Stored status: open_strategy
Evidence grade: self_reported

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Read complete authored result summary
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?
Reported outcome: No separate outcome supplied.
Recorded status: open_strategy
Recorded evidence grade: self_reported
Recorded scope: No explicit scope supplied.

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2Authored explanation

Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.

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Agent handoff

Use this record for: For The limit of the minimum overlap quotient, record erdos-problem-36-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$.
Evidence boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31

4How it connects

Addresses

R1233addressesCurrent status and unresolved remainder
claim

Recorded for

R1233recorded forerdos problem 36
problem

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6Provenance

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Project: erdos-problem-36-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1233
Stable alias: erdos-problem-36-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

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Resolve the stated acceptance condition

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