partition relations
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2021 ◽  
pp. 1-18
Author(s):  
Mirna Džamonja ◽  
Angeliki Koutsoukou-Argyraki ◽  
Lawrence C. Paulson
Keyword(s):  


2021 ◽  
Vol 9 ◽  
Author(s):  
Assaf Rinot ◽  
Jing Zhang

Abstract We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $\kappa $ , if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $ , there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$ , there exist $x\neq y$ in X such that $f(x+y)=g$ .



2019 ◽  
Vol 85 (1) ◽  
pp. 87-102
Author(s):  
NATASHA DOBRINEN ◽  
DANIEL HATHAWAY

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.



2019 ◽  
Vol 20 (01) ◽  
pp. 1950015
Author(s):  
Jing Zhang

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height [Formula: see text] has a nonspecial subtree of size [Formula: see text]. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of [Formula: see text], which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations.



2019 ◽  
Vol 259 ◽  
pp. 50-66
Author(s):  
Alan Dow
Keyword(s):  


2019 ◽  
Vol 244 (2) ◽  
pp. 147-166
Author(s):  
Diana Ojeda-Aristizabal ◽  
William Weiss
Keyword(s):  


2018 ◽  
Vol 225 (2) ◽  
pp. 771-796 ◽  
Author(s):  
Dilip Raghavan ◽  
Stevo Todorcevic
Keyword(s):  


Author(s):  
Joanna Jureczko

AbstractThe first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.



2017 ◽  
Vol 82 (4) ◽  
pp. 1560-1575 ◽  
Author(s):  
NATASHA DOBRINEN ◽  
DAN HATHAWAY

AbstractSeveral variants of the Halpern–Läuchli Theorem for trees of uncountable height are investigated. Forκweakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finited≥ 2, we prove the consistency of the Halpern–Läuchli Theorem ondmany normalκ-trees at a measurable cardinalκ, given the consistency of aκ+d-strong cardinal. This follows from a more general consistency result at measurableκ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.



2017 ◽  
Vol 44 (10) ◽  
pp. 721-731 ◽  
Author(s):  
Phillip B. Drain ◽  
Brian J. Monaghan ◽  
Guangqing Zhang ◽  
Raymond. J. Longbottom ◽  
Michael W. Chapman ◽  
...  


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