## Combinatorics of numbers |

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### Contents

Filters and ultrafilters | 10 |

4 Semigroup of ultrafilters | 26 |

7 Van der Waerden Theorem | 40 |

Copyright | |

2 other sections not shown

### Common terms and phrases

21 is m-regular arbitrarily long arithmetic arbitrary element arithmetic progressions called cardinality choose a subset Choose an arbitrary closed subsemigroup closed subset combinatorical compact space Consider an arbitrary contains arbitrarily long continuous map converges Corollary countable set Denote discrete space equation exists a finite exists a natural exists a subset exists an infinite family 21 family of subsets filter finite number finite subset K C G FP-set free semigroup free ultrafilter Furstenberg-Weiss Theorem group G Hales-Jewett Theorem Hausdorff space Hindman Theorem idempotent infinite set infinite version Lemma long arithmetic progressions m-regular with respect minimal right ideal monochromatic solution natural numbers neighborhood nonempty number of subsets open sets open subset Orb(x Proof Prove the following Ramsey Theorem ring Schur Theorem semigroup 3(N set of natural space 3X subset H Suppose theorem for partitions topological space uniformly recurrent Waerden Theorem word f(x