# Erdős cardinal
An **Erdős cardinal** is a large cardinal defined by a partition property: for an ordinal α at least ω, the α-th Erdős cardinal (also called a partition cardinal), written κ(α), is the least cardinal κ such that every coloring of the finite subsets of κ into two colors has a homogeneous subset of κ of order type α, the relation κ → (α)²<ω[1](https://link.springer.com/article/10.1007/s40879-018-0278-1). The name honors [Paul Erdős](https://www.edgepedia.org/paul-erdos), whose work with [András Hajnal](https://www.edgepedia.org/andras-hajnal) and [Richard Rado](https://www.edgepedia.org/richard-rado) founded the partition calculus, the branch of combinatorial set theory from which these cardinals arose[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf). The case α = ω₁ is the one most often meant by "Erdős cardinal" in the literature[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf).
| Key fact | Detail |
|---|---|
| Definition | κ(α) is the least cardinal κ with κ → (α)²<ω: every 2-coloring of the finite subsets of κ has a homogeneous subset of order type α[1](https://link.springer.com/article/10.1007/s40879-018-0278-1) |
| ZFC facts | If κ(α) exists it is regular; if α is a limit ordinal, κ(α) is strongly inaccessible, so V_κ is a model of ZFC[3](https://d-nb.info/1171637721/34) |
| Provability | By Gödel's incompleteness theorem, ZFC cannot prove that κ(α) exists[3](https://d-nb.info/1171637721/34) |
| Indiscernibles | κ → (α)<ω holds exactly when every structure in a countable language with universe containing κ has a set of indiscernibles of order type α[4](https://www.math.uni-hamburg.de/spag/ml/MLSeminar/Slides/bagaria.introduction.pdf) |
| Relation to L | The ω₁^L-Erdős cardinal does not exist in the constructible universe L, so its existence implies V ≠ L[4](https://www.math.uni-hamburg.de/spag/ml/MLSeminar/Slides/bagaria.introduction.pdf) |
| Hierarchy position | A Ramsey cardinal satisfies κ → (κ)<ω and is exactly the κ-Erdős cardinal; Erdős cardinals for smaller α sit below Ramsey[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf) • [5](https://neugierde.github.io/cantors-attic/Ramsey) |
| Key case | κ(ω₁), the least κ with κ → (ω₁)²<ω, is the cardinal commonly called the Erdős cardinal[1](https://link.springer.com/article/10.1007/s40879-018-0278-1) • [2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf) |
## Definition and basic properties
The partition relation κ → (α)²<ω says that whenever the finite subsets of κ are partitioned into two classes, some subset of κ of order type α has all its finite subsets in one class. [Ramsey's theorem](https://www.edgepedia.org/ramseys-theorem) for infinite sets is the special case ω → (ω)²₂[1](https://link.springer.com/article/10.1007/s40879-018-0278-1). For each α ≥ ω, the least cardinal with this property, if there is one, is denoted κ(α) and called the α-th Erdős cardinal or partition cardinal[1](https://link.springer.com/article/10.1007/s40879-018-0278-1).
Two facts hold in ZFC alone. First, κ(α), if it exists, is regular. Second, when α is a limit ordinal, κ = κ(α) is strongly inaccessible, which means V_κ is a model of ZFC; by Gödel's incompleteness theorem, ZFC therefore cannot deduce the existence of κ(α)[3](https://d-nb.info/1171637721/34). Anyone who needs such a cardinal must add its existence as a new axiom of infinity[3](https://d-nb.info/1171637721/34).
## Indiscernibles
The partition property has a model-theoretic equivalent that explains its usefulness. For α ≥ ω, the relation κ → (α)<ω holds if and only if every structure M in a countable language whose universe includes κ has a set X ⊆ κ of order type α consisting of indiscernibles for M: for every formula φ(x₁, …, xₙ) of the language and every two increasing n-tuples from X, φ holds of one tuple exactly when it holds of the other[4](https://www.math.uni-hamburg.de/spag/ml/MLSeminar/Slides/bagaria.introduction.pdf). A homogeneous set for a coloring is automatically a set of indiscernibles for any structure in a countable language, since each formula defines a color class on n-tuples[4](https://www.math.uni-hamburg.de/spag/ml/MLSeminar/Slides/bagaria.introduction.pdf).
This formulation connects Erdős cardinals to the theory of indiscernibles, the technique behind Silver's analysis of 0^# and the indiscernibles for L. The existence of 0^# can itself be used as a large-cardinal assumption, and it lies below the existence of the Erdős cardinal in strength[1](https://link.springer.com/article/10.1007/s40879-018-0278-1).
## Place in the large cardinal hierarchy
Erdős cardinals generalize Ramsey cardinals. Erdős and Hajnal's first joint paper, from 1958, proved that a measurable cardinal satisfies κ → (κ)<ω, and this relation defines the Ramsey cardinals[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf). A [Ramsey cardinal](https://www.edgepedia.org/ramsey-cardinal) is exactly the κ-Erdős cardinal for α = κ[5](https://neugierde.github.io/cantors-attic/Ramsey). For smaller α the corresponding cardinals are weaker: κ(ω₁) sits below the Ramsey cardinals, and the consequences of κ → (ω₁)<ω for descriptive set theory found in the mid-1960s by Frederick Rowbottom, Jack Silver, and Robert Solovay were what made the weaker relation important[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf). In recognition of Erdős's role in defining large cardinals through partition relations, the cardinals satisfying κ → (ω₁)<ω came to be called Erdős cardinals[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf).
The cardinals are genuinely large, not implied by the classical hierarchy below them. Erdős and Hajnal proved in 1962 that the least inaccessible cardinal fails κ → (ℵ₁)²<ω, so the least inaccessible cardinal is not an Erdős cardinal[6](https://people.clas.ufl.edu/jal/files/jal30g.pdf). Above the ordinary Erdős cardinals, Cantor's Attic records that below any Ramsey cardinal there are stationarily many completely ineffable, greatly Erdős cardinals[5](https://neugierde.github.io/cantors-attic/Ramsey), and the "greatly Erdős" property has been used to calibrate strengthenings of the Chang property, one of which is equiconsistent with a Ramsey cardinal[7](https://www.sciencedirect.com/science/article/pii/S0168007211000571).
## Consequences for the constructible universe
The existence of an Erdős cardinal contradicts the minimal universe L. A consequence of the existence of κ(ω₁^L), drawn from work of Rowbottom and Silver, is that the ω₁^L-Erdős cardinal does not exist in L, so the existence of such a cardinal implies V ≠ L[4](https://www.math.uni-hamburg.de/spag/ml/MLSeminar/Slides/bagaria.introduction.pdf). The same failure appears in core model theory: Menachem Magidor, professor at the Einstein Institute of Mathematics of the [Hebrew University of Jerusalem](https://www.edgepedia.org/hebrew-university-of-jerusalem), proved in 1990 that if there is no inner model with an Erdős cardinal, that is, no κ with κ → (ω₁)<ω, then for every ordinal β the core model K carries an algebra on β with countably many operations such that every subset of β closed under those operations is a countable union of sets in K[8](https://cris.huji.ac.il/en/publications/representing-sets-of-ordinals-as-countable-unions-of-sets-in-the-/). William Mitchell, of the [University of Florida](https://www.edgepedia.org/university-of-florida), developed the corresponding positive theory in his 1997 work on Jónsson cardinals, Erdős cardinals, and the core model, studying δ-Erdős cardinals, in which the set of indiscernibles is required only to have ordertype δ for a regular cardinal δ < κ[9](https://arxiv.org/abs/math/9706207).
The ω-Erdős cardinal also calibrates newer hypotheses. Trevor Wilson, of the Department of Mathematics at [Miami University](https://www.edgepedia.org/miami-university), proved in 2019 that a non-Σ₂-reflecting weakly remarkable cardinal is equiconsistent with the existence of an ω-Erdős cardinal, and that the generic Vopěnka principle with "Ord is not Δ₂-Mahlo" is equiconsistent with a proper class of ω-Erdős cardinals[10](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/258571CBD89531A93CB2681F46E9EF23/S0022481218000762a.pdf/weakly-remarkable-cardinals-erds-cardinals-and-the-generic-vopenka-principle.pdf).
## History
The partition calculus began with Erdős and Rado's 1956 paper "A partition calculus in set theory", the first work of major scope aiming at a calculus of partitions[11](https://mathshistory.st-andrews.ac.uk/Extras/Hajnal_books/). The 1965 paper "Partition relations for cardinal numbers" by Erdős, Hajnal, and Rado, published in *Acta Mathematica Academiae Scientiarum Hungaricae* 16, pages 93–196, gave an almost complete discussion of the ordinary partition relation for cardinals under the generalized continuum hypothesis[11](https://mathshistory.st-andrews.ac.uk/Extras/Hajnal_books/) • [12](https://portal.mardi4nfdi.de/wiki/Publication%3A5541367). The 1984 book *Combinatorial set theory: partition relations for cardinals* by Erdős, Hajnal, Attila Máté, and Rado collected the combinatorial core of the subject, including the ordinary partition relation without that hypothesis[13](https://real-eod.mtak.hu/15973/). The Erdős cardinals themselves emerged from this program as the large cardinals most directly expressible in partition notation, with κ(ω₁) as the canonical example[2](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf).
## References
1. [N. H. Bingham and A. J. Ostaszewski, "Set theory and the analyst," European Journal of Mathematics (2018)](https://link.springer.com/article/10.1007/s40879-018-0278-1)
2. ["The Mathematics of Paul Erdős," Notices of the AMS, vol. 45, no. 1 (1998)](https://www.ams.org/journals/notices/199801/199801FullIssue.pdf)
3. [N. H. Bingham and A. J. Ostaszewski, "Set theory and the analyst" (full PDF)](https://d-nb.info/1171637721/34)
4. [Joan Bagaria, "Introduction to large cardinals," seminar slides, Universitat de Barcelona](https://www.math.uni-hamburg.de/spag/ml/MLSeminar/Slides/bagaria.introduction.pdf)
5. ["Ramsey cardinal," Cantor's Attic](https://neugierde.github.io/cantors-attic/Ramsey)
6. [J. M. Larson, "Erdős–Hajnal partition calculus" (chapter draft, University of Florida)](https://people.clas.ufl.edu/jal/files/jal30g.pdf)
7. ["Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties," Annals of Pure and Applied Logic 162 (2011), 863–902](https://www.sciencedirect.com/science/article/pii/S0168007211000571)
8. [Menachem Magidor, "Representing sets of ordinals as countable unions of sets in the core model," Transactions of the American Mathematical Society 317 (1990), 91–126](https://cris.huji.ac.il/en/publications/representing-sets-of-ordinals-as-countable-unions-of-sets-in-the-/)
9. [William Mitchell, "Jónsson Cardinals, Erdős Cardinals, and the Core Model," arXiv:math/9706207](https://arxiv.org/abs/math/9706207)
10. [Trevor M. Wilson, "Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle," Journal of Symbolic Logic 84 (2019), 1711–1721](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/258571CBD89531A93CB2681F46E9EF23/S0022481218000762a.pdf/weakly-remarkable-cardinals-erds-cardinals-and-the-generic-vopenka-principle.pdf)
11. ["Hajnal books," MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Hajnal_books/)
12. ["Partition relations for cardinal numbers," MaRDI portal publication record](https://portal.mardi4nfdi.de/wiki/Publication%3A5541367)
13. [Erdős, Hajnal, Máté, and Rado, "Combinatorial set theory. Partition relations for cardinals," Akadémiai Kiadó, Budapest, 1984](https://real-eod.mtak.hu/15973/)
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