# P3010: The 3-uniform partition relation for the continuum

- ID: `P3010`
- Reference: `erdos-problem-70`
- Page: https://theoremdb.org/statements/P3010
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ holds.

### Context

This is Problem 70 from the Erdős problems database, concerning partition calculus in set theory. The 3-uniform case is the triple analogue of the classical ordinal-cardinal Ramsey property. The cases $n \leq 3$ are known to hold trivially, so the first non-trivial case is $n = 4$. Erdős and Rado proved that $\mathfrak{c} \to (\omega + n, 4)^3_2$ for all finite $n \geq 2$, leaving open whether the result extends to $\beta = \omega \cdot 2$ with $n = 4$.

### Problem setup

- **Definition (The cardinality of the continuum, denoted $\mathfrak{c}$).** The cardinality of the continuum, denoted $\mathfrak{c}$, is the cardinality of the set of real numbers.
- **Definition (An ordinal $\beta$).** An ordinal $\beta$ is countable if its cardinality is at most $\aleph_0$, the cardinality of the natural numbers.
- **Definition (For an ordinal $\alpha$, the order type of a subset $s \subseteq \alpha$).** For an ordinal $\alpha$, the order type of a subset $s \subseteq \alpha$ is the unique ordinal order-isomorphic to $s$ with the induced ordering.
- **Definition (A 2-coloring of 3-element subsets of $\alpha$).** A 2-coloring of 3-element subsets of $\alpha$ is a function assigning each unordered 3-element subset to one of two colors (conventionally red and blue), formally encoded as a symmetric predicate on ordered triples of distinct elements.
- **Definition (A subset).** A subset is red-monochromatic of order type $\beta$ if it has order type $\beta$ and every 3-element subset is colored red.
- **Definition (A subset).** A subset is blue-monochromatic of cardinality $c$ if it has cardinality $c$ and every 3-element subset is colored blue.
- **Remark.** This is Problem 70 from the Erdős problems database, concerning partition calculus in set theory. The 3-uniform case is the triple analogue of the classical ordinal-cardinal Ramsey property. The cases $n \leq 3$ are known to hold trivially, so the first non-trivial case is $n = 4$. Erdős and Rado proved that $\mathfrak{c} \to (\omega + n, 4)^3_2$ for all finite $n \geq 2$, leaving open whether the result extends to $\beta = \omega \cdot 2$ with $n = 4$.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ holds.

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 70 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ 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 70 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ holds.

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 70 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 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ holds.

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 70 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 70 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 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ holds. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-70`, 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 70, maintained status record. Erdős Problems record 70, checked 2026-08-01. Problem 70; status field and linked bibliography https://www.erdosproblems.com/70
   - Also cited at Problem 70; 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 The 3-uniform partition relation for the continuum: 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 70 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 The 3-uniform partition relation for the continuum: 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 70. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/70.lean:L77; theorem erdos_70; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/70.lean#L77
   - 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 The 3-uniform partition relation for the continuum: Supplies the pinned formal declaration whose human-readable editorial statement is published here.