Some results in the partition calculus.

1999 ◽  
Author(s):  
Albin Lee Jones
Keyword(s):  
Partition Calculus

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.

A Theorem in the Partition Calculus Corrigendum

1974 ◽  
Vol 17 (2) ◽  
pp. 305-305 ◽  
Author(s):  
P. Erdös ◽  
E. C. Milner
Keyword(s):  
Partition Calculus

Some problems in the partition calculus

1971 ◽  
pp. 183-185
Author(s):  
Richard Rado
Keyword(s):  
Partition Calculus

scholarly journals Countable chain condition in partition calculus

1998 ◽  
Vol 188 (1-3) ◽  
pp. 205-223 ◽  
Author(s):  
Stevo Todorcevic
Keyword(s):  
Chain Condition ◽  
Partition Calculus ◽  
Countable Chain Condition ◽  
Countable Chain

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

More consistency results in partition calculus

1993 ◽  
Vol 81 (1-2) ◽  
pp. 97-110 ◽  
Author(s):  
Saharon Shelah ◽  
Lee Stanley
Keyword(s):  
Partition Calculus ◽  
Consistency Results

Edwin W. Miller. On a property of families of sets. English with Polish summary. Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego (Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie), Class III, vol. 30 (1937), pp. 31–38. - Ben Dushnik and Miller E. W.. Partially ordered sets. American journal of mathematics, vol. 63 (1941), pp. 600–610. - P. Erdős. Some set-theoretical properties of graphs. Revista, Universidad Nacional de Tucumán, Serie A, Matemáticas y física teórica, vol. 3 (1942), pp. 363–367. - G. Fodor. Proof of a conjecture of P. Erdős. Acta scientiarum mathematicarum, vol. 14 no. 4 (1952), pp. 219–227. - P. Erdős and Rado R.. A partition calculus in set theory. Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–489. - P. Erdős and Rado R.. Intersection theorems for systems of sets. The journal of the London Mathematical Society, vol. 35 (1960), pp. 85–90. - A. Hajnal. Some results and problems on set theory. Acta mathematica Academiae Scientiarum Hungaricae, vol. 11 (1960), pp. 277–298. - P. Erdős and Hajnal A.. On a property of families of sets. Acta mathematica Academiae Scientiarum Hungaricae, vol. 12 (1961), pp. 87–123. - A. Hajnal. Proof of a conjecture of S. Ruziewicz. Fundamenta mathematicae, vol. 50 (1961), pp. 123–128. - P. Erdős, Hajnal A. and Rado R.. Partition relations for cardinal numbers. Acta mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196. - P. Erdős and Hajnal A.. On a problem of B. Jónsson. Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, vol. 14 (1966), pp. 19–23. - P. Erdős and Hajnal A.. On chromatic number of graphs and set-systems. Acta mathematica Academiae Scientiarum Hungaricae, vol. 17 (1966), pp. 61–99.

1995 ◽  
Vol 60 (2) ◽  
pp. 698-701
Author(s):  
James E. Baumgartner
Keyword(s):  
Set Theory ◽  
American Mathematical Society ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Class Iii ◽  
Ordered Sets ◽  
Partition Calculus ◽  
Cardinal Numbers ◽  
Set Systems ◽  
Partition Relations
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