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

- ID: `P2956`
- Reference: `erdos-problem-1`
- Page: https://theoremdb.org/statements/P2956
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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 $1$.

### Problem setup

- **Definition (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.
- **Remark.** 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 $1$.

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-1`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 1, maintained status record. Erdős Problems record 1, checked 2026-08-01. Problem 1; status field and linked bibliography https://www.erdosproblems.com/1
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 1 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>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 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/1.lean#L115
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.