[#P2956] Erdős's sum-distinct set problem for real numbers

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Problem. Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq |\sum_{a \in S_1} a - \sum_{a \in S_2} a|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{|A|} < N$ holds.

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Definitions and notation

1Context

This is a generalization of Erdős's classical problem on sum-distinct sets of integers to the real number setting. The integer version asks whether $N \gg 2^n$ for a sum-distinct set $A \subseteq \{1, \ldots, N\}$ with $|A| = n$, where all subset sums are distinct. The real version relaxes the distinctness condition to require only that subset sums differ by at least $1$.

2Definitions

Definition 1 (A finite set of real numbers $A$). A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and for any two distinct subsets $S_1, S_2 \subseteq A$, the inequality $1 \leq |\sum_{a \in S_1} a - \sum_{a \in S_2} a|$ holds.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq |\sum_{a \in S_1} a - \sum_{a \in S_2} a|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{|A|} < N$ holds.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 1 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq |\sum_{a \in S_1} a - \sum_{a \in S_2} a|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{|A|} < N$ holds.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 1 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq \|\sum_{a \in S_1} a - \sum_{a \in S_2} a\|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{\|A\|} < N$ holds.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 1 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq \|\sum_{a \in S_1} a - \sum_{a \in S_2} a\|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{\|A\|} < N$ holds.^[1] Relevance to this problem For erdos problem 1, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 1 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A. 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 1 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 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq \|\sum_{a \in S_1} a - \sum_{a \in S_2} a\|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{\|A\|} < N$ holds. Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq \|\sum_{a \in S_1} a - \sum_{a \in S_2} a\|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{\|A\|} < N$ holds.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq \|\sum_{a \in S_1} a - \sum_{a \in S_2} a\|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{\|A\|} < N$ holds.^[1] Relevance to this problem For Erdős's sum-distinct set problem for real numbers, record erdos-problem-1-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 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. 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 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq \|\sum_{a \in S_1} a - \sum_{a \in S_2} a\|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \cdot 2^{\|A\|} < N$ holds. 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 material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 1 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 1 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 connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemErdős's sum-distinct set problem for real numbers

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“Erdős's sum-distinct set problem for real numbers.” TheoremDB. P2956. Problem statement; statement text SHA-256 fe368e2fab3fbb6da5e18f76618ac1c158bf5afccb64910452984aeb96dcbe3a. https://theoremdb.org/statement/?ref=P2956

BibTeX

@misc{theoremdb-problem-fe368e2fab3fbb6da5e18f76618ac1c158bf5afccb64910452984aeb96dcbe3a,
  title = {{Erdős's sum-distinct set problem for real numbers}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 fe368e2fab3fbb6da5e18f76618ac1c158bf5afccb64910452984aeb96dcbe3a},
  url = {https://theoremdb.org/statement/?ref=P2956}
}

Plain text: Built Markdown snapshot

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 1, maintained status record. Erdős Problems record 1, checked 2026-08-01. Problem 1; 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 1; 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 sum-distinct set problem for real numbers: 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 1 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 sum-distinct set problem for real numbers: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 1. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/1.lean:L115; theorem erdos_1.variants.real; 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 sum-distinct set problem for real numbers: 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.

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[#P2956] Erdős's sum-distinct set problem for real numbers

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Background

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