# Erdős-Rado theorem

Repeated exponentiation^{} for cardinals is denoted ${\mathrm{exp}}_{i}(\kappa )$, where $$. It is defined by:

$${\mathrm{exp}}_{0}(\kappa )=\kappa $$ |

and

$${\mathrm{exp}}_{i+1}(\kappa )={2}^{{\mathrm{exp}}_{i}(\kappa )}$$ |

The Erdős-Rado theorem states that:

$${\mathrm{exp}}_{i}{(\kappa )}^{+}\to {({\kappa}^{+})}_{\kappa}^{i+1}$$ |

That is, if $f:{[{\mathrm{exp}}_{i}{(\kappa )}^{+}]}^{i+1}\to \kappa $ then there is a homogeneous set of size ${\kappa}^{+}$.

As special cases, ${({2}^{\kappa})}^{+}\to {({\kappa}^{+})}_{\kappa}^{2}$ and ${({2}^{{\mathrm{\aleph}}_{0}})}^{+}\to {({\mathrm{\aleph}}_{1})}_{{\mathrm{\aleph}}_{0}}^{2}$.

Title | Erdős-Rado theorem |
---|---|

Canonical name | ErdHosRadoTheorem |

Date of creation | 2013-03-22 12:59:30 |

Last modified on | 2013-03-22 12:59:30 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 9 |

Author | Henry (455) |

Entry type | Theorem |

Classification | msc 05D10 |

Classification | msc 03E05 |

Related topic | arrowsrelation |

Related topic | ArrowsRelation |