[#P3010] The 3-uniform partition relation for the continuum

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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.

A finite mathematical diagram showing two-colored triples on a finite vertex set. — Two-colored triples on a finite vertex set.

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Definitions and notation

1Context

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 $n = 4$.

2Definitions

Definition 1 (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 2 (An ordinal $\beta$). An ordinal $\beta$ is countable if its cardinality is at most $\aleph_0$, the cardinality of the natural numbers.

Definition 3 (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 4 (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 5 (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 6 (A subset). A subset is blue-monochromatic of cardinality $c$ if it has cardinality $c$ and every 3-element subset is colored blue.

3What 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.

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 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]

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Record	Kind	Assessment
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.	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 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] Relevance to this problem For erdos problem 70, pins the dated research frontier: 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. 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 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. Open full record →
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.	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 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] Relevance to this problem For The 3-uniform partition relation for the continuum, record erdos-problem-70-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 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. 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 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. 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 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.

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ProblemThe 3-uniform partition relation for the continuum

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Bases over one fixed field and the Axiom of Choicemathematical logic
Does strong jump traceability imply LR reducibility?mathematical logic
A random-preserving partial injection that lowers Turing degreemathematical logic

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“The 3-uniform partition relation for the continuum.” TheoremDB. P3010. Problem statement; statement text SHA-256 786f0fb0b97462827b18b6dc7cab5b917647d37f1982f95d0305ab99afff0064. https://theoremdb.org/statement/?ref=P3010

BibTeX

@misc{theoremdb-problem-786f0fb0b97462827b18b6dc7cab5b917647d37f1982f95d0305ab99afff0064,
  title = {{The 3-uniform partition relation for the continuum}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 786f0fb0b97462827b18b6dc7cab5b917647d37f1982f95d0305ab99afff0064},
  url = {https://theoremdb.org/statement/?ref=P3010}
}

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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 70, maintained status record. Erdős Problems record 70, checked 2026-08-01. Problem 70; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 70 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 The 3-uniform partition relation for the continuum: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 70. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/70.lean:L77; theorem erdos_70; 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 The 3-uniform partition relation for the continuum: 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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[#P3010] The 3-uniform partition relation for the continuum

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