共查询到20条相似文献,搜索用时 31 毫秒
1.
We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ,λ, denoted by (κ,<λ)⇝P, of the classical relation (κ,n,λ)→ρ in infinite combinatorics. By definition, (κ,n,λ)→ρ holds if every map F: [κ]
n
→[κ]<λ
has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ,n,λ)→n+1 holds iff κ ≥ λ
+n
, where λ
+n
denotes the n-th cardinal successor of an infinite cardinal λ. By using the (κ,<λ)⇝P framework, we present a self-contained proof of the first author’s result that (λ
+n
,n,λ)→n+2, for each infinite cardinal λ and each positive integer n, which solves a problem stated in the 1985 monograph of Erdős, Hajnal, Máté, and Rado. Furthermore, by using an order-dimension
estimate established in 1971 by Hajnal and Spencer, we prove the relation $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1}
{2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor }
$(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1}
{2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor }
, for every infinite cardinal λ and all positive integers n and r with 2≤r<n. For example, (ℵ210,4,ℵ0)→32,768. Other order-dimension estimates yield relations such as (ℵ109,4,ℵ0) → 257 (using an estimate by Füredi and Kahn) and (ℵ7,4,ℵ0)→10 (using an exact estimate by Dushnik). 相似文献
2.
Notes on combinatorial set theory 总被引:1,自引:0,他引:1
Saharon Shelah 《Israel Journal of Mathematics》1973,14(3):262-277
We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when ℵα is regular, │ξ│+<ωα. (2) There is a Jonsson algebra in ℵα+n, and \aleph _{a + n} \not \to \left[ {\aleph _{a + n} } \right]_{\aleph _{a + n} }^{n + 1} if 2^{\aleph _{ - - } } = \aleph
_{a + n} \cdot (3) If λ>ℵ0 is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (ℵα regular) │f(x)∩f(y│<ℵα, then there is a free subset of order-type ζ for every ζ<ωα+1. 相似文献
3.
We extend a transitive model V of ZFC+GCH cardinal preservingly to a model N of ZF + “GCH holds below ℵ ω ” + “there is a surjection from the power set of ℵ ω onto λ”, where λ is an arbitrarily high fixed cardinal in V. The construction can be described as follows: add ℵ n +1 many Cohen subsets of ℵ n+1 for every n < ω, and adjoin λ many subsets of ℵ ω which are unions of ω-sequences of those Cohen subsets; then let N be a choiceless submodel generated by equivalence classes of the λ subsets of ℵ ω modulo an appropriate equivalence relation. 相似文献
4.
Jean-Pierre Levinski Menachem Magidor Saharon Shelah 《Israel Journal of Mathematics》1990,69(2):161-172
We establish, starting from some assumptions of the order of magnitude of a huge cardinal, the consistency of (ℵω+1,ℵω)↠(ω1,ω0), as well as of some other transfer properties of the type (κ+,κ)↠(α+,α), where κ is singular. 相似文献
5.
Saharon Shelah 《Israel Journal of Mathematics》1996,95(1):61-114
We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the
cofinality of (S
≤ℵ
0(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ+,2) in more cases even when cf(λ)=ℵ0 (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure
I had already written but did not find (Section 6).
Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by
the Basic Research Fund, Israel Academy of Sciences. Publication number 430. 相似文献
6.
Saharon Shelah 《Israel Journal of Mathematics》1987,59(3):299-326
We give bounds for
where cfδ=ℵ1, (∀a<δ)
, in cases which previously remained opened, including the first such cardinal: theω
1-th cardinal inC
ω=∩n<ω
C
n whereC
0 is the cardinal andC
n+1 the set of fixed points ofC
n. No knowledge of earlier results is required. A subsequent work generalizing this was applied to many more cardinals ([Sh
7]).
The author would like to thank the Canadian NSERC for supporting this research by Grant A3040 and the Israel Academy of Science
for supporting it. 相似文献
7.
Saharon Shelah 《Central European Journal of Mathematics》2010,8(2):213-234
Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing
ℵ1,ℵ2 by λ, λ
+ (starting with λ = λ
<λ
> ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ
+ but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c. 相似文献
8.
Stephen H. Hechler 《Israel Journal of Mathematics》1973,15(4):384-395
For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN
κ of κ copies ofN (the discrete space of cardinality ℵ0). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is
consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal
less than or equal to 2ℵ
0. Mrówka has also defined and studied the class ℳ={κ: Exp (N
κ)=κ} whereN
κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot
even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2ℵ
0 is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which
are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2ℵ
0 and that ℳ nevertheless contain all cardinals no greater than 2ℵ
0. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any
normal measure on a measurable cardinal, then μ(ℳ)=0. 相似文献
9.
Let b denote the unboundedness number of ωω. That is, b is the smallest cardinality of a subset
such that for everyg∈ωω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that
, and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ0, ℵ1, ℵ1, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ0, ℵ0, λ, 1〉, andB is not well-generated, then λ≥b. 相似文献
10.
A. Hajnal 《Combinatorica》1985,5(2):137-139
We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG
0,G
1 with χ(G
0)=χ(G
1)=ϰ+ and χ(G
0×G
1)=ϰ. We also prove a result from the other direction. If χ(G
0)≧≧ℵ0 and χ(G
1)=k<ω, then χ(G
0×G
1)=k. 相似文献
11.
Jean-Pierre Levinski 《Israel Journal of Mathematics》1984,48(2-3):225-243
We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that
if ρ is a cardinal such that ρ ≧ ω1, and (ρ++,ρ+↠(ρ+, ρ), then 0+ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries,
we get:
Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧
K
+ ≧ω2 and (λ+, λ)↠(K
+,K. Then 0+ exists.
Theorem B.Assume that there exists a singularcardinal κ such that(K
+,K↠(ω1, ω0. Then 0+ exists.
Theorem C.Assume that (λ
++, λ). Then 0+ exists (also ifK=ω
0.
Remark. Here, as in the paper of Donder and Koepke, “O+ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model.
Theorem D.Assume that (λ
++, λ)↠(K
+, K) and thatK≧ω
1. Then 0+ exists.
Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω
2, or that κ is inaccessible. We shall recall this in due place.
Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K
+, K↠(ωn +1, ωn), for κ singular andn≧0. 0266 0152 V 3
The consistency of several statements of this sort—a prototype of which is (N
ω+1,N
ω)↠(ω1, ω0) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker
than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the
author of the present paper. 相似文献
12.
Arthur W. Apter 《Israel Journal of Mathematics》1985,52(4):273-292
Letκ be a 3 huge cardinal in a countable modelV of ZFC, and letA andB be subsets of the successor ordinals <κ so thatA ⋃B={α<κ:α is a successor ordinal}. Using techniques of Gitik, we construct a choiceless modelN
A of ZF of heightκ so thatN
A ╞“ZF+⌍AC
ω+Forα ∈A, ℵa is a Ramsey cardinal+Forβ ∈B, ℵβ is a singular Rowbottom cardinal which carries a Rowbottom filter+Forγ a limit ordinal, ℵy is a Jonsson cardinal which carries a Jonsson filter”.
The author wishes to express his thanks to the Rutgers Research Council for a Summer Research Fellowship which partially supported
this work.
The author also wishes to thank Moti Gitik and Bob Mignone for their useful comments concerning the subject matter of this
paper. 相似文献
13.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
14.
We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the
p-rank of Ext
ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable
strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π0 and Π1, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2μ = μ+ such that the p-rank of Ext
ℤ(G, ℤ) equals 2μ = μ+ for every p ∈ Π0 and 0 otherwise, that is, for p ∈ Π1.
Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development
project No. I-706-54.6/2001.
Supported by a grant from the German Research Foundation DFG.
__________
Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007. 相似文献
15.
Saharon Shelah 《Israel Journal of Mathematics》2012,191(2):507-543
We force 2 λ to be large, and for many pairs in the interval (λ, 2 λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)+3, hL(X) ≤ s(X)+3 and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ 0 we get hd(X), hL(X) ≤ N 2. 相似文献
16.
The Grunsky coefficient inequalities play a crucial role in various problems and are intrinsically connected with the integrable
holomorphic quadratic differentials having only zeros of even order. For the functions with quasi-conformal extensions, the
Grunsky constant ℵ(f) and the extremal dilatationk(f) are related by ℵ(f)≤k(f). In 1985, Jürgen Moser conjectured that any univalent functionf(z)=z+b
0+b
1
z
−1+… on Δ*={|z|>1} can be approximated locally uniformly by functions with ℵ(f)<k(f). In this paper, we prove a theorem confirming Moser’s conjecture, which sheds new light on the features of Grunsky coefficients.
In memory of Jürgen Moser
The research was supported by the RiP program of the Volkswagen-Stiftung in the Mathematisches Forschungsinstitut Oberwolfach. 相似文献
17.
V. M. Korchevsky 《Vestnik St. Petersburg University: Mathematics》2011,44(4):268-271
New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables
X
1, X
2, …, with finite variances are established. No particular type of dependence between the random variables in the sequence
is assumed. The statement of the theorem involves the classical condition Σ
n
∞ (log2
n)2/n
2 < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence
condition. 相似文献
18.
We consider different generalizations of Martin’s Axiom to higher cardinals. For ℵ1, assuming CH+2ℵ
1>ℵ2+□ℵ1 we show that a generalized Martin’s Axiom considered by Baumgartner settles the ℵ2 Souslin Hypothesis ... the wrong way. We further show that, assuming CH+2ℵ
1>ℵ2, a strengthening of this axiom implies □ℵ
1. Finally, we show that a seemingly innocuous further strengthening is inconsistent with CH+2ℵ
1>ℵ2.
This author thanks the US-Israel Binational Science Foundation for partial support of this research. 相似文献
19.
We show that that if every real has a sharp and there are Δ
2
1
-definable prewellorderings of ℝ of ordinal ranks unbounded inω
2, then there is an inner model for a strong cardinal. Similarly, assuming the same sharps, the Core ModelK is Σ
3
1
-absolute unless there is an inner model for a strong cardinal. 相似文献
20.
Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR
m
(ε) of constant sectional curvature ε satisfies a basic inequality δ(n
1,…,n
k
)≤c(n
1,…,n
k
)H
2+b(n
1,…,n
k
)ε, whereH is the mean curvature of the immersion, andc(n
1,…,n
k
) andb(n
1,…,n
k
) are constants depending only onn
1,…,n
k
,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n
1,…,n
k
). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n
1+…+n
k
+1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n
1+…+n
k
+2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms.
The author was supported by the NSFC and RFDP. 相似文献