Partition relations for uncountable ordinals

1975 ◽  
Vol 21 (4) ◽  
pp. 296-307 ◽  
Author(s):  
James E. Baumgartner
Keyword(s):  
Partition Relations

NEGATIVE PARTITION RELATIONS OR UNCOUNTABLE CARDINALS OF COFINALITY

2001 ◽  
Vol 37 (1-2) ◽  
pp. 233-236
Author(s):  
P. Matet
Keyword(s):  
Partition Relations ◽  
Uncountable Cardinals

We modify an argument of Baumgartner to show that…

Partition Relations

2009 ◽  
pp. 129-213 ◽  
Author(s):  
András Hajnal ◽  
Jean A. Larson
Keyword(s):  
Partition Relations

Partition Relations for Cardinal Numbers

2013 ◽  
pp. 119-125
Author(s):  
Hans Jürgen Prömel
Keyword(s):  
Cardinal Numbers ◽  
Partition Relations

Model theory under the axiom of determinateness

1985 ◽  
Vol 50 (3) ◽  
pp. 773-780
Author(s):  
Mitchell Spector
Keyword(s):  
Model Theory ◽  
Axiom Of Choice ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Innovation ◽  
First Order ◽  
Partition Relations ◽  
Hanf Number ◽  
The Axiom Of Choice

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

On Finite Polarized Partition Relations

1969 ◽  
Vol 12 (3) ◽  
pp. 321-326 ◽  
Author(s):  
V. Chvátal
Keyword(s):  
Cardinal Numbers ◽  
Partition Relations ◽  
Image Position

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write(1)if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is writtenand means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

scholarly journals FORCING AND THE HALPERN–LÄUCHLI THEOREM

2019 ◽  
Vol 85 (1) ◽  
pp. 87-102
Author(s):  
NATASHA DOBRINEN ◽  
DANIEL HATHAWAY
Keyword(s):  
Partition Relations ◽  
Finite Products

AbstractWe investigate the effects of various forcings on several forms of the Halpern– Läuchli theorem. For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern– Läuchli theorem at κ. We also show that the Halpern–Läuchli theorem is preserved by <κ-closed forcings assuming κ is measurable, following some observed reflection properties.

scholarly journals Partition relations for successor cardinals

1986 ◽  
Vol 59 (2) ◽  
pp. 152-169 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Partition Relations

Weak compactness and square bracket partition relations

1972 ◽  
Vol 37 (4) ◽  
pp. 673-676 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Weak Compactness ◽  
Partition Relation ◽  
Tree Property ◽  
Weakly Compact ◽  
Partition Relations

Although there are many characterizations of weakly compact cardinals (e.g. in terms of indescnbability and tree properties as well as compactness) the most interesting set-theoretic (combinatorial) one is in terms of partition relations. To be more precise we define for κ and α cardinals and n an integer the partition relation of Erdös, Hajnal and Rado [2] as follows:For every function F: [κ]n→ α (called a partition of [κ]n, the n-element subsets of κ, into α pieces), there exists a set C⊆ κ (called homogeneous for F) such that card C = κ and F″[C]n≠ α, i.e. some element of the range is omitted when F is restricted to the n-element subsets of C. It is the simplest (nontrivial) of these relations, i.e. , that is the well-known equivalent of weak compactness.1Two directions of inquiry immediately suggest themselves when weak compactness is described in terms of these partition relations: (a) Trying to strengthen the relation by increasing the superscript—e.g., —and (b) trying to weaken the relation by increasing the subscript—e.g., . As it turns out, the strengthening to is only illusory for using the equivalence of to the tree property one quickly sees that implies (and so is equivalent to) for every n. Thus is the strongest of these partition relations. The second question seems much more difficult.

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