Pierre Matet, Saharon Shelah
Let kappa be a regular uncountable cardinal and lambda > kappa a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of P_kappa (lambda) and use this to prove that the nonstationary ideal on P_kappa (lambda) is nowhere precipitous.
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We prove that if cf(lambda) > aleph_0 and 2^{cf(lambda)}<lambda, then lambda->(lambda,omega+1)^2.
Suppose lambda is a singular cardinal of uncountable cofinality kappa. For a model M of cardinality lambda, let No(M) denote the number of isomorphism types of models N of cardinality lambda which are L_{infty lambda}-equivalent to M. In [Sh:189] inverse kappa-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A) were considered. It was proved that for every cardinal mu there exists an inverse kappa-system A such that A consists of abelian groups having cardinality at most mu^kappa and card(Gr(A)/Fact(A))= mu. In [Sh:228] a strict connection between inverse kappa-…
We are proving the following: (1) If $\kap$ is a weakly inaccessible then $NS_\kap$ is not $\kap^+$-saturated. (2) If $\kap$ is a weakly inaccessible and $\tet <\kap$ is regular then $NS^\tet_\kap$ is not $\kap^+$-saturated. (3) If $\kap$ is singular then $NS^{cf\kap}_{\kap^+}$ is not $\kap^{++}$-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If $\kap >\aleph_1$ then $NS_\kap$ is not $\kap^+$-saturated. (B) If $\tet^+<\kap$ then $NS^\tet_\kap$ is not $\kap^+$-saturated.
Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
A cardinal lambda is called omega-inaccessible if for all mu < lambda we have mu^omega<lambda. We show that for every omega-inaccessible cardinal lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelof regular space of density lambda. This extends an analogous earlier result of ours that only worked for regular lambda.
In their paper from 1981, Milner and Sauer conjectured that for any poset P, if cf(P)=lambda>cf(lambda)=kappa, then P must contain an antichain of size kappa. We prove that for lambda>cf(lambda)=kappa, if there exists a cardinal mu<lambda such that cov(lambda,mu,kappa,2)=lambda, then any poset of cofinality lambda contains lambda^kappa antichains of size kappa. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.
We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambd…
We extend a transitive model V of ZFC + GCH cardinal preservingly to a model N of ZF + "GCH holds below Alef_omega" + "there is a surjection from the power set of Alef_omega onto lambda" where lambda is an arbitrarily high fixed cardinal in V.
We prove the consistency of $\mathfrak{r}_λ<\mathfrak{d}_λ$ and even $\mathfrak{u}_λ<\mathfrak{d}_λ$ for a singular cardinal $λ$.
We prove that if lambda is a strong limit singular cardinal and kappa a regular uncountable cardinal < lambda, then NS_{kappa lambda}, the non-stationary ideal over P_{kappa} lambda, is nowhere precipitous. We also show that under the same hypothesis every stationary subset of P_{kappa} lambda can be partitioned into lambda^{< kappa} disjoint stationary sets.
The main result is that for lambda strong limit singular failing the continuum hypothesis (i.e. 2^lambda > lambda^+), a polarized partition theorem holds.
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length epsilon in lambda. We prove for a large set of successor cardinals lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality lambda. We concentrate on successors of singular cardinals.
1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no (incomparability-)antichain of cardinality lambda, while L^{n+1} has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality lambda_n, but not one of cardinality lambda_n^+ .
This note gives two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular, while the second is a negative consistency result which applies to the situation of lambda singular. Both results are connected to an earlier result of the same authors in which they showed that a certain version of clubsuit holds at a successor of singular just in ZFC. The first result here shows that that fact can to a certain extent be extended to the successor of singular. The negative result here gives limitations to the extent to which one …
We present two different types of models where, for certain singular cardinals lambda of uncountable cofinality, lambda -> (lambda, omega+1)^2, although lambda is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, that, for example, consistently, aleph_{omega_1} not-> (aleph_{omega_1}, omega+1)^2 and consistently, 2^{aleph_0} not-> (2^{aleph_0},omega +1)^2 .
In this short note we show that if lambda>aleph_1 is regular and lambda is not the successor of a singular cardinal of cofinality aleph_0, and G is a lambda-free abelian group of size lambda, then there is a free group G' subseteq G of size lambda.
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
Let B(kappa, lambda) be the subalgebra of P(kappa) generated by [kappa]^{<= lambda}. It is shown that if B is any homomorphic image of B(kappa, lambda) then either |B|< 2^lambda or |B|=|B|^lambda, moreover if X is the Stone space of B then either |X| <= 2^{2^lambda} or |X|=|B|=|B|^lambda. This implies the existence of 0-dimensional compact T_2 spaces whose cardinality and weight spectra omit lots of singular cardinals of ``small'' cofinality.
We prove that on many inaccessible there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal lambda which is lambda x omega-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra on the successor of beth_omega. Lastly, we give further information on guessing of clubs.
Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [CK], modulo the assumption of supercompactness. In section 2 we construct several lambda-Archimedean ultrapowers of omega under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exi…
We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of B_1 and of B_2 than kappa. We give a condition (for lambda, mu and theta) which implies that for some BA A_theta there are B_1=B^1_{lambda, mu, theta} and B_2B^2_{lambda, mu, theta} such that Depth (B_t) <= mu and Depth (B_1 oplus_{A_theta} B_1) >= lambda. We then investigate for a fixed A, the existence of such B_1,B_2 giving sufficient and necessary condi…
Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Puttin…
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | i < kappa) of regular cardinals converging to mu such that lambda = tcf(prod_{i < kappa} lambda_i, <_{J^{bd}_kappa}). 2. Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them has an uncountable cofinality. Then there is sigma_* < mu such that for every chi < theta the following holds: theta > …
It is proved that every singular cardinal $λ$ admits a function $RTS:[λ^+]^2\rightarrow[λ^+]^2$ that transforms rectangles into squares. Namely, for every cofinal subsets $A,B$ of $λ^+$, there exists a cofinal subset $C$ of $lambda^+$, such that $RTS[AxB]$ covers CxC. When combined with a recent result of Eisworth, this shows that Shelah's notion of strong coloring $Pr_1(λ^+,λ^+,λ^+,\cf(λ))$ coincides with the classical negative partition relation $λ^+\not\rightarrow[λ^+]^2_{λ^+}$.
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