# P2996: Existence of asymptotically optimal Sidon sets in initial intervals

- ID: `P2996`
- Reference: `erdos-problem-44`
- Page: https://theoremdb.org/statements/P2996
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 44: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. For a real number $\varepsilon > 0$, determine whether the following holds: for all sufficiently large natural numbers $M$, there exists a Sidon set $A \subseteq \{1, 2, \ldots, M\}$ such that $|A| \geq (1 - \varepsilon)\sqrt{M}$.

### Context

This is a variant of Erdős Problem 44, concerning the construction of large Sidon sets in initial intervals of the natural numbers without requiring extension of a given starting set.

### Problem setup

- **Definition (A Sidon set).** A Sidon set is a set of natural numbers in which all sums of two (not necessarily distinct) elements are distinct.
- **Remark.** This is a variant of Erdős Problem 44, concerning the construction of large Sidon sets in initial intervals of the natural numbers without requiring extension of a given starting set.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 44: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. For a real number $\varepsilon > 0$, determine whether the following holds: for all sufficiently large natural numbers $M$, there exists a Sidon set $A \subseteq \{1, 2, \ldots, M\}$ such that $|A| \geq (1 - \varepsilon)\sqrt{M}$.

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 44 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 44: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. For a real number $\varepsilon > 0$, determine whether the following holds: for all sufficiently large natural numbers $M$, there exists a Sidon set $A \subseteq \{1, 2, \ldots, M\}$ such that $|A| \geq (1 - \varepsilon)\sqrt{M}$. [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 44 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 44: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. For a real number $\varepsilon > 0$, determine whether the following holds: for all sufficiently large natural numbers $M$, there exists a Sidon set $A \subseteq \{1, 2, \ldots, M\}$ such that $|A| \geq (1 - \varepsilon)\sqrt{M}$.

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 44 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 44: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. For a real number $\varepsilon > 0$, determine whether the following holds: for all sufficiently large natural numbers $M$, there exists a Sidon set $A \subseteq \{1, 2, \ldots, M\}$ such that $|A| \geq (1 - \varepsilon)\sqrt{M}$.

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 44 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 44 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 44: A set $A$ of natural numbers is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ in $A$ are distinct. For a real number $\varepsilon > 0$, determine whether the following holds: for all sufficiently large natural numbers $M$, there exists a Sidon set $A \subseteq \{1, 2, \ldots, M\}$ such that $|A| \geq (1 - \varepsilon)\sqrt{M}$. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-44`, 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 44, maintained status record. Erdős Problems record 44, checked 2026-08-01. Problem 44; status field and linked bibliography https://www.erdosproblems.com/44
   - Also cited at Problem 44; 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 Existence of asymptotically optimal Sidon sets in initial intervals: 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 44 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 Existence of asymptotically optimal Sidon sets in initial intervals: 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 44. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/44.lean:L54; theorem erdos_44.variants.empty_start; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/44.lean#L54
   - 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 Existence of asymptotically optimal Sidon sets in initial intervals: Supplies the pinned formal declaration whose human-readable editorial statement is published here.