A counter-example in the partition calculus for an uncountable ordinal

1980 ◽  
Vol 36 (3-4) ◽  
pp. 287-299 ◽  
Author(s):  
Jean A. Larson
Keyword(s):  
Partition Calculus ◽  
Counter Example

Some remarks on the partition calculus of ordinals

1999 ◽  
Vol 64 (2) ◽  
pp. 436-442 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Regular Cardinal ◽  
Continuum Hypothesis ◽  
Alternative Proof ◽  
Partition Relation ◽  
Negative Relation ◽  
Partition Calculus ◽  
Counter Example ◽  
The Continuum

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

Variations on the Theme of Scarf's Counter-Example

2001 ◽  
Author(s):  
Alok Kumar ◽  
Martin Shubik
Keyword(s):  
Counter Example

scholarly journals Roberts’ weak welfarism theorem: a minor correction

2021 ◽  
Author(s):  
Peter J. Hammond
Keyword(s):  
Euclidean Space ◽  
Social Welfare ◽  
Utility Function ◽  
Weak Continuity ◽  
Strict Preference ◽  
Counter Example ◽  
Independence Of Irrelevant Alternatives ◽  
Unrestricted Domain ◽  
A Minor ◽  
Minor Correction

AbstractRoberts’ “weak neutrality” or “weak welfarism” theorem concerns Sen social welfare functionals which are defined on an unrestricted domain of utility function profiles and satisfy independence of irrelevant alternatives, the Pareto condition, and a form of weak continuity. Roberts (Rev Econ Stud 47(2):421–439, 1980) claimed that the induced welfare ordering on social states has a one-way representation by a continuous, monotonic real-valued welfare function defined on the Euclidean space of interpersonal utility vectors—that is, an increase in this welfare function is sufficient, but may not be necessary, for social strict preference. A counter-example shows that weak continuity is insufficient; a minor strengthening to pairwise continuity is proposed instead and its sufficiency demonstrated.

scholarly journals The Annihilation of Space: A Bad (Historical) Concept

2021 ◽  
pp. 1-30
Author(s):  
Alexis D. Litvine
Keyword(s):  
Cultural History ◽  
Intellectual History ◽  
Social History ◽  
Economic Consequences ◽  
Cultural Mediation ◽  
Counter Example ◽  
Dead End ◽  
The Social ◽  
History Of ◽  
Representations Of Space

Abstract This article is a reminder that the concept of ‘annihilation of space’ or ‘spatial compression’, often used as a shorthand for referring to the cultural or economic consequences of industrial mobility, has a long intellectual history. The concept thus comes loaded with a specific outlook on the experience of modernity, which is – I argue – unsuitable for any cultural or social history of space. This article outlines the etymology of the concept and shows: first, that the historical phenomena it pretends to describe are too complex for such a simplistic signpost; and, second, that the term is never a neutral descriptor but always an engagement with a form of historical and cultural mediation on the nature of modernity in relation to space. In both cases this term obfuscates more than it reveals. As a counter-example, I look at the effect of the railways on popular representations of space and conclude that postmodern geography is a relative dead end for historians interested in the social and cultural history of space.

Non-maximal and disconnected stability groups: SU(3) counter-example to Michel conjecture

1985 ◽  
Vol 154 (2-3) ◽  
pp. 159-165 ◽  
Author(s):  
J. Burzlaff ◽  
T. Murphy ◽  
L. O'Raifeartaigh
Keyword(s):  
Counter Example

Conflict-free access of arrays — a counter example

1980 ◽  
Vol 10 (1) ◽  
pp. 20 ◽  
Author(s):  
Ashoke Deb
Keyword(s):  
Free Access ◽  
Counter Example
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