arXiv subjects
Publications and source records attributed to Mirna Džamonja.
If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.
In this paper we investigate some properties of first order theories which prevent them from having universal models under certain cardinal arithmetic assumptions. Our results give a new syntactical condition, oak property, which is a sufficient condition for a theory not to have universal models in cardinality lambda when certain cardinal arithmetic assumptions implying the failure of GCH (and close to the failure of SCH) hold.
The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/9504221) and [CuSh:596] (math.LO/9703219). While various ZFC restrictions on the one hand, and independence results on the other, were discovered about the weak reflection, the question of the relative consistency of the existence of a regular cardinal kappa such that the first cardinal weakly reflecting at kappa is a successor of singular, remained open. This paper answers that question by proving that (modulo large cardinal assumptions close to 2-hugeness) that there indeed can be such a cardinal kappa.
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=lambda^+ and lambda is a strong limit singular, then the corresponding guessing holds. Both results are also connected to an earlier result of Dzamonja and Shelah 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 a result of math.LO/9601219 can to a certain extent be extended to the successor of a regular. The negative result here gives limitations to the extent to which one can hope to extend this result.
Various versions of club are shown to be different. A question of Soukup, Fuchino and Juhasz, is it consistent to have a stick without club, is answered as a consequence. The more detailed version of the paper, which is coming up, also answers a question of Galvin.
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 can hope to extend the afore mentioned fact. This note is presently only available as an electronic publication. It can be obtained as a ps file either from the e-prints archive, or from the authors' home pages.
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
Suppose that lambda = mu^+. We consider two aspects of the square property on subsets of lambda. First, we have results which show e.g. that for aleph_0 <= kappa =cf (kappa)< mu, the equality cf([mu]^{<= kappa}, subseteq)= mu is a sufficient condition for the set of elements of lambda whose cofinality is bounded by kappa, to be split into the union of mu sets with squares. Secondly, we introduce a certain weak version of the square property and prove that if mu is a strong limit, then this weak square property holds on lambda without any additional assumptions
In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that for all alpha < theta with cf(alpha)= lambda there is C club in alpha such that F restriction C is strictly increasing. We prove that it is consistent to have a cardinal theta such that strong non-reflection and weak reflection each hold on an unbounded set of cardinals less than theta .
We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure is equivalent to the existence of a real-valued measurable cardinal less or equal to c. We show that not being in MS is preserved by all forcing extensions which do not collapse omega_1, while being in MS can be destroyed even by a ccc forcing.