A theorem and some consistency results in partition calculus

Shelah, Saharon; Stanley, Lee J.

doi:10.1016/0168-0072(87)90015-7

Cited by 20 publications

(20 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1.3) Parallels with [SSI,§2]. For the most part the material in §2 of this paper is a straightforward adaptation to a recursive morass construction of the treatment of §2 of [SSI], except, of course that there is no need to formally develop historical conditions since for a < co2 the sequences ((bß, cß) : ß < a) are rather special kinds of historical conditions.…”

Section: Preliminariesmentioning

confidence: 99%

“…The first occurs in (2.3), in the proof of the lemma of (2.2). There, Velleman has simplified and improved the argument of [SSI,(2.15.4)], eliminating the need for the auxiliary partition there. This builds on a more thorough analysis of the histories of points, not limited to the finite set of stages when the history is nontrivial, as in [SSI] (though the latter notion continues to be of central importance here).…”

Section: Preliminariesmentioning

confidence: 99%

“…As usual, we construct this function from small pieces which are assigned to the levels of the morass. The definition of these small pieces is motivated by the definition of Pbasic in [SSI,§2].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [SSI,§5], it was shown that Con(ZFC + 3 a weakly compact cardinal) => Con( ZFC + 2N' = N2 + V/c < co(co3cox -> (co3cox, k)1) ). In the extension, 2N° = min((2N°),/, N2) ; full GCH can be obtained or violated in the usual ways, as desired.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

$\omega\sb 3\omega\sb 1\to (\omega\sb 3\omega\sb 1,3)\sp 2$ requires an inaccessible

Stanley¹,

Velleman²,

Morgan³

1991

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract.We show that if there is a simplified (a>2, 1 )-morass with linear limits and 2 ' = N2 , then wiiax ■** (oJia>x, 3) . Thus, assuming 2 ' = N2 , this negative relation holds in V if both N2 and N3 are (successor cardinals) , since in this case, well-known arguments show there is a simplified (ca2, 1)-morass with linear limits . The contrapositive is that, assuming 2 ' = N2 , the positive relation holds only if either N2 or N3 is (inaccessible) .

show abstract

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

$\omega\sb 3\omega\sb 1\to (\omega\sb 3\omega\sb 1,3)\sp 2$ requires an inaccessible

Stanley¹,

Velleman²,

Morgan³

1991

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The construction of the partial order uses historical forcing [3]. In this method conditions are allowed into the partial order if they can be constructed from the amalgamation of simpler conditions satisfying certain properties.…”

Section: The Partial Ordermentioning

confidence: 99%

Identities on cardinals less than ℵ_ω

Gilchrist

Shelah

1996

J. symb. log.

View full text Add to dashboard Cite

Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on vertices are colored with ℵ0 colors. In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [2]. When the graph being colored has size ℵ1, the identities that must occur have been classified by Shelah [4]. If the graph has size greater than or equal to ℵω the identities have also been classified in [3]. The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to situations where more colors are used.There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on vertices. First consider the branches in the complete binary tree of height ω to be vertices of a complete graph.

show abstract