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A partition theorem for ordinals

1979
*
Journal of combinatorial theory. Series A
*

In this note we show that oi -Gans (

doi:10.1016/0097-3165(79)90030-x
fatcat:hmxcmql3arh47kktrbadrd7m54
*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 [3]*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

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 Milner (unpublished) generalized his result to prove the following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*[8] , stated below as

*Theorem*2.6, yields:

*For*all natural numbers m, 09 -+ (09, m) 2 . ...

##
###
An order property of partition cardinals

1970
*
Bulletin of the Australian Mathematical Society
*

This note studies cardinal numbers K which have

doi:10.1017/s0004972700045834
fatcat:fyijunszyzcdtgcd5zmhjgfbbe
*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]

2022
*
arXiv
*
pre-print

In fact, no such equivalence relation makes

arXiv:2209.09765v1
fatcat:ufca33farzarrbdjuy6goqrbua
*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]

2022
*
arXiv
*
pre-print

In fact, no such equivalence relation makes

arXiv:2112.04980v5
fatcat:xggskfqawrgd5nzt5fpi3cy6sm
*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

1972
*
Pacific Journal of Mathematics
*

Let w be

doi:10.2140/pjm.1972.43.1
fatcat:o63adpjcrndgropidhhq4erhoa
*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]

2004
*
arXiv
*
pre-print

defined

arXiv:math/0404014v1
fatcat:w5ufiitj7jgyfcyeqi3zsgmy4m
*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]

2019
*
arXiv
*
pre-print

This paper considers three different

arXiv:1904.07790v1
fatcat:d2z5xcrc5nfxrf7a2mkgtbi6om
*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 [2]*for*the full proof. ...##
###
Block combinatorics

2006
*
Transactions of the American Mathematical Society
*

Results contain (

doi:10.1090/s0002-9947-06-03864-5
fatcat:kwadwhoazvgovcbi7k4s55ksma
*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]

2004
*
arXiv
*
pre-print

Results contain (

arXiv:math/0406188v1
fatcat:gkr423skh5fjrj3mu4pvivqhy4
*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

1994
*
Fundamenta Mathematicae
*

Pol has shown that

doi:10.4064/fm-144-3-243-258
fatcat:jzgci6d2qnbfxgyim745w24eym
*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

1990
*
Proceedings of the American Mathematical Society
*

This provides

doi:10.2307/2047989
fatcat:lupk5lbc3jcjjhnot2gme7zhly
*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

1990
*
Proceedings of the American Mathematical Society
*

This provides

doi:10.1090/s0002-9939-1990-1007520-3
fatcat:bakvsmwzezboncpm7267vyowiq
*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

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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