[#P3036] Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure

A finite mathematical diagram showing an infinite-pattern sample and an affine copy on the real line. — A finite pattern and an affine image on the real line.

Problem. Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?

1Context

This problem belongs to the intersection of combinatorial geometry and measure theory, asking whether infinite subsets of the real line can always be excluded from some set of positive measure under all non-degenerate affine transformations.

2Problem setup

Definition 1 (Definition 1). For a set $A \subseteq \mathbb{R}$, a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist real numbers $a$ and $b$ with $a \neq 0$ such that the image of $A$ under the map $x \mapsto a \cdot x + b$ is contained in $E$.

Definition 2 (The Lebesgue measure of a measurable set $E \subseteq \mathbb{R}$). The Lebesgue measure of a measurable set $E \subseteq \mathbb{R}$ is denoted $\text{volume}(E)$; the set has positive measure if $0 < \text{volume}(E)$.

Remark 1. This problem belongs to the intersection of combinatorial geometry and measure theory, asking whether infinite subsets of the real line can always be excluded from some set of positive measure under all non-degenerate affine transformations.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?

1Status

Current status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?^[1]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?^[1] Relevance to this problem For erdos problem 120, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$? Open full record →
	Trace	Reported
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?^[1] Relevance to this problem For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure, record erdos-problem-120-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 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion materialContext, examples, and computations

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 120 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connectTyped relations and evidence flow

ProblemErdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Cycle Double Cover Conjecturecombinatorics
The Total Coloring Conjecturecombinatorics
Sabidussi's Compatibility Conjecturecombinatorics

How to cite

TheoremDB contributors, “Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-120

This page as plain text: erdos-problem-120.md

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 120, maintained status record. Erdős Problems record 120, checked 2026-08-01. Problem 120; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 120; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 120 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 120. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/120.lean:L45; theorem erdos_120; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P3036] Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance