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### A partition theorem for ordinals

Diana Schmidt
1979 Journal of combinatorial theory. Series A
In this note we show that oi -Gans (a, m)" for all m < w and every (Y of the form UJ~, where Q: +trans (a, m)* is a weakening of the usyal partitiop property obtained by considering only partitions whose  ...  tirst member is a transitive relation.  ...  Erdijs and Rado first asked in  for which ordinals 01 and which cardinals m the partition property a? + (01, ,)i holds.  ...

### A short proof of a partition theorem for the ordinal ωω

Jean A. Larson
1973 Annals of Mathematical Logic
For any ordinal a, one writes a -~ (a, m) 2 if and only if for any set A order-isomorphic to a, and any function f from the pairs of elements of A into {0, 1}, either there is a subset X c_ A order-isomorphic  ...  Milner (unpublished) generalized his result to prove the following theorem: For all natural numbers m, w'o -* (co'o , m) 2 .  ...  Larson, A short proof of a partition theorem for the ordinal tot° Ramsey's Theorem  , stated below as Theorem 2.6, yields: For all natural numbers m, 09 -+ (09, m) 2 .  ...

### An order property of partition cardinals

N. H. Williams
1970 Bulletin of the Australian Mathematical Society
This note studies cardinal numbers K which have a partition property which amounts to the following. Let V be a cardinal, H an ordinal limit number and m a positive integer.  ...  having in their i-th place (for i = 1, ..., m) a subset of H. which contains exactly n. elements.  ...  The following theorem provides for their existence. THEOREM 4 . 4 Let v be a cardinal and n a limit ordinal. SupposeK is a cardinal such that < •* (r\.m) w . Then K -*• (r\) u .  ...

### Characterizations of ordinal analysis [article]

James Walsh
2022 arXiv   pre-print
In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.  ...  First, we characterize ordinal analysis as a partition of Σ^1_1-definable and Π^1_1-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal  ...  Ordinal Analysis as a Partition Before diving into the proof of Theorem 1.4, we should check that the ordinal analysis partition is a good partition.  ...

### A characterization of ordinal analysis [article]

James Walsh
2022 arXiv   pre-print
In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.  ...  Ordinal analysis induces a partition of Σ^1_1-definable and Π^1_1-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal.  ...  Our main theorem is that no good partition makes a single distinction that the ordinal analysis partition does not make. Theorem 1.5. Let " be good.  ...

### The use of mitotic ordinals in cardinal arithmetic

Alexander Abian
1972 Pacific Journal of Mathematics
Let w be a mitotic ordinal. Then w is a limit ordinal. Moreover, for every element Si of a mitotic partition (Si) iew of w we have: (1) U Si = sup S { = w . Proof.  ...  A nonzero ordinal w is called mitotic if and only if it can be partitioned into W pairwise disjoint subsets each of type w. Such a partition is called a mitotic partition of w.  ...

### Ramsey and Nash-Williams combinatorics via Schreier families [article]

Vassiliki Farmaki
2004 arXiv   pre-print
defined for every countable ordinal ξ.  ...  The main results of this paper (a) extend the finite Ramsey partition theorem, and (b) employ this extension to obtain a stronger form of the infinite Nash-Williams partition theorem, and also a new proof  ...  The derivation of Ellentuck's theorem We finally show that our Theorem 3.10(= Theorem B ′ ) implies, using the simple argument contained in Theorem 4.6, Ellentuck's theorem (and hence, Galvin-Prikry's  ...

### A Comparison of Pair and Triple Partition Relations [article]

Jonathon Eric Beers
2019 arXiv   pre-print
This paper considers three different partition relations from partition calculus, two of which are pair relations and one of which is a triple relation.  ...  or negative triple partition relations.  ...  Theorem 1 (Balanced Erdős-Rado Partition Relation). For all infinite cardinals κ and for all nonempty ordinals γ < cf(κ), (2 <κ ) + → (κ + 1) 2 γ . Proof. Refer to  for the full proof.  ...

### Block combinatorics

V. Farmaki, S. Negrepontis
2006 Transactions of the American Mathematical Society
Results contain (a) a block partition Ramsey theorem for every countable ordinal ξ (Hindman's Theorem corresponding to ξ = 1, and the Milliken-Taylor Theorem to ξ a finite ordinal), (b) a countable ordinal  ...  form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.  ...  Thanks are due to Ted Odell and Haskell Rosenthal for helpful comments and discussions on the content of this paper and also to the Department of Mathematics of the University of Texas at Austin for the  ...

### Block Combinatorics [article]

V. Farmaki, S. Negrepontis
2004 arXiv   pre-print
Results contain (a) a block partition Ramsey theorem for every countable ordinal xi (Hindman's theorem corresponding to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable ordinal form  ...  of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.  ...  Acknowledgments The authors wish to thank Ted Odell and Haskell Rosenthal for helpful comments and discussions on the content of this paper.  ...

### Universal spaces in the theory of transfinite dimension, I

Wojciech Olszewski
1994 Fundamenta Mathematicae
Pol has shown that for every countable ordinal α, there exists a universal space for separable metrizable spaces X with ind X = α.  ...  We prove that for every countable limit ordinal λ, there is no universal space for separable metrizable spaces X with Ind X = λ.  ...  Observe that K contains a partition in S β × I n = S β+n between distinct points of the base B β+n for all but a finite number of ordinals β < λ.  ...

### Page 2995 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews
{Reviewer’s remark: The theorem quoted in the first paragraph of this review holds not only for partitions of [w]* but also for Borel (and slightly more complicated) partitions of [w]®; see a paper by  ...  Instead, we quote a corollary that combines and extends partition theorems of Ramsey and van der Waerden. Let [w}? be partitioned into finitely many pieces.  ...

### Strong Preinjective Partitions and Representation Type of Artinian Rings

Birge Zimmermann-Huisgen
1990 Proceedings of the American Mathematical Society
This provides a very elementary proof for Auslander's theorem saying that for Artin algebras vanishing of the left pure global dimension is equivalent to finiteness of the representation type.  ...  be grouped to a unique "strong preinjective partition" while the finitely presented right modules possess a "strong preprojective partition"; these strong partitions are upgraded versions of the partitions  ...  Zimmermann for several helpful conversations.  ...

### Strong preinjective partitions and representation type of Artinian rings

Birge Zimmermann-Huisgen
1990 Proceedings of the American Mathematical Society
This provides a very elementary proof for Auslander's theorem saying that for Artin algebras vanishing of the left pure global dimension is equivalent to finiteness of the representation type.  ...  be grouped to a unique "strong preinjective partition" while the finitely presented right modules possess a "strong preprojective partition"; these strong partitions are upgraded versions of the partitions  ...  Zimmermann for several helpful conversations.  ...

### Subspaces of ordinals

Valentin Gutev
2014 Topology and its Applications
a r t i c l e i n f o a b s t r a c t MSC: 54F05 Keywords: Orderable space Suborderable space Ordinal space It was recently proved that each subspace of an ordinal space is also orderable.  ...  The present note aims to give a simple proof of this fact.  ...  The proof now can be accomplished along the lines of the idea of [3, Theorem 4.2] for rearranging partitions of orderable subspaces, see Theorem 3.1.  ...
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