# P2980: Unbounded representation function for sets with cofinite sumset

- ID: `P2980`
- Reference: `erdos-problem-28`
- Page: https://theoremdb.org/statements/P2980
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.

### Context

This problem originates from Erdős's collection of open problems in additive combinatorics. It concerns the behavior of representation functions for sets with cofinite sumsets.

### Problem setup

- **Definition (The sumset $A + A$ of a set $A \subseteq \mathbb{N}$).** The sumset $A + A$ of a set $A \subseteq \mathbb{N}$ is the set $\{a + b : a, b \in A\}$ of all sums of two (not necessarily distinct) elements of $A$.
- **Definition (The representation function $r_A(n)$ counts the number of ordered pairs $(a, b) \in A \times A$ with $a + b = n$).** The representation function $r_A(n)$ counts the number of ordered pairs $(a, b) \in A \times A$ with $a + b = n$.
- **Definition (A set $S \subseteq \mathbb{N}$).** A set $S \subseteq \mathbb{N}$ is cofinite if its complement $S^c = \mathbb{N} \setminus S$ is finite.
- **Remark.** This problem originates from Erdős's collection of open problems in additive combinatorics. It concerns the behavior of representation functions for sets with cofinite sumsets.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 28 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$. [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 28 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 28 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 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$.

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 28 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 28 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 28: Let $A$ be a set of natural numbers. For each $n \in \mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \{a + b : a, b \in A\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\limsup_{n \to \infty} r_A(n) = \infty$. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-28`, 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 28, maintained status record. Erdős Problems record 28, checked 2026-08-01. Problem 28; status field and linked bibliography https://www.erdosproblems.com/28
   - Also cited at Problem 28; 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 Unbounded representation function for sets with cofinite sumset: 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 28 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 Unbounded representation function for sets with cofinite sumset: 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 28. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/28.lean:L36; theorem erdos_28; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/28.lean#L36
   - 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 Unbounded representation function for sets with cofinite sumset: Supplies the pinned formal declaration whose human-readable editorial statement is published here.