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

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

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.

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 $n = 4$.

2Problem setup

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.

Remark 1. 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 $n = 4$.

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

Current 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]

1Packet records

Record	Kind	Assessment
	Result	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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.

How the 2 records connectTyped relations and evidence flow

ProblemThe 3-uniform partition relation for the continuum

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Erdős Problem 152: isolated sums in Sidon setserdos problems
Erdős's sum-distinct set problem for real numberserdos problems
Erdős's Conjecture on Divergent Reciprocal Sums and Arithmetic Progressionserdos problems

How to cite

TheoremDB contributors, “The 3-uniform partition relation for the continuum,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-70

This page as plain text: erdos-problem-70.md

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.

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

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance