arXiv subjects
Publications and source records attributed to Shimon Garti.
We define Magidor cardinals as Jónsson cardinals upon replacing colorings of finite subsets by colorings of $\aleph_0$-bounded subsets. Unlike Jónsson cardinals which appear at some low level of large cardinals, we prove the consistency of having quite large cardinals along with the fact that no Magidor cardinal exists.
We prove that if there is a real-valued measurable cardinal then the splitting number is $\aleph_1$. Likewise, if the continuum is real-valued measurable then the reaping number equals the continuum.
Let $κ$ be any regular cardinal. Assuming the existence of a huge cardinal above $κ$, we prove the consistency of $\binom{κ^{++}}{κ^+}\rightarrow\binomτ{κ^+}$ for every ordinal $τ<κ^{++}$. Likewise, we prove a full amenable relation for $(\aleph_2,\aleph_1)$ with respect to collections which are strongly closed under countable intersections.
We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filters are beyond ZFC-inconsistency
We prove that the Devlin-Shelah weak diamond implies Galvin's property. On the other hand, Galvin's property is consistent with the negation of the weak diamond, and even with Martin's axiom. We show that the proper forcing axiom implies a relative to the negation of Galvin's property for $\aleph_1$.
We show that the character spectrum $Sp_χ(λ)$, for a singular cardinal $λ$ of countable cofinality, may include any prescribed set of regular cardinals between $λ$ and $2^λ$.
We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $μ$ one can force the strong polarized relation with respect to the pair $μ^+,μ$.
We prove the existence of infinite dense free sets (in the usual topology) for set mappings on the reals, under reasonable assumptions.
We define separating properties for normal ultrafilters. We prove that compactness and supercompactness are separable, yet compactness and measurability are not. We describe how to use separating properties in order to elicit distinct normal ultrafilters which do not contain the measurables.
We prove that the strong polarized relation of $θ$ above $ω$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal invariant on the continuum. We show that similar results hold for a supercompact cardinal, and for a strong limit singular under some assumptions.
We characterize the situation of having many normal measures on a measurable cardinal. We show the plausibility of having many normal measures on each compact cardinal.
Let V be a finite set of points in the plane, not contained in a line. Assume |V| = n is an odd number, and |L \cap V| \leq 3 for every line L which is spanned by V. We prove that every simple line L_{a,b} in V creates a simple wedge (i.e., a triple {a, b, c} \subseteq V such that L_{a,b} and L_{a,c} are simple lines). We also show that both restrictions on V (namely |V| is odd and |L \cap V| \leq 3) are needed. We conjecture, further, that if |V | = n is an odd number then V contains a simple wedge, even if V is not 3-bounded. We introduce a method for proving this, which gives (in this paper) partial results.
We define the strong splitting number, prove that it equals s when exists, and put some restrictions on the possibility that s is a singular carcinal.
We prove the consistency of a strong polarized relation for a cardinal and its successor, using pcf and forcing
We prove the consistency of a singular cardinal $λ$ with small value of the ultrafilter number $u_λ$, and arbitrarily large value of $2^λ$.
We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.
We prove that the strong polarized relation for the continuum holds for $\aleph_0$ and for every supercompact cardinal. We use iteration of Mathias forcing.
We generalize Rothberger's theorem for regular lambda. We prove that the discrepancy between the pseudointersection number and the towering number is limited, under some reasobable assumption