arXiv subjects
Publications and source records attributed to Shimon Garti.
We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property and explores stronger forms of this property in the context of large cardinals
Jing Zhang proved the consistency of $\binom{ω_2}{ω_1}\rightarrow\binom{n}{ω_1}_ω$ for every $n\inω$ with the negative relation $\binom{ω_2}{ω_1}\nrightarrow\binomω{ω_1}_ω$. We reduce the consistency strength of this statement to an $ω_1$-Erdos cardinal.
We suggest a forcing version of Yablo's paradox and discuss its implication on self-reference.
We prove the consistency of $\binom{μ^+}μ\nrightarrow\binom{μ^+ ω_1}{μ μ}$ where $μ$ is a strong limit singular cardinal of countable cofinality. This result can be forced at limit of measurable cardinals and at small cardinals like $\aleph_ω$.
We study the structured rainbow Ramsey theory at uncountable cardinals. When compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain structural constraints, such as being stationary or closed in its supremum. In the process of dealing with cardinals greater than $ω_1$, we uncover some connections between versions of Chang's Conjectures and instances of rainbow Ramsey partition relations, addressing a question raised in \cite{zhang}.
Erdos, Hajnal and Rado asked whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binomμ{\aleph_2}\rightarrow\binomμ{\aleph_2}_2$ is independent over ZF for some $μ$ of cofinality $ω_1$.
We prove that if $μ>{\rm cf}(μ)=ω$ and $2^μ=μ^+$ then $\binom{μ^+}μ\nrightarrow\binom{μ^+\ \aleph_2}{μ μ}$.
We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $λ$ for which the sequence $\langle 2^θ:θ<λ\rangle$ is not eventually constant and the weak diamond fails at $λ$. We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible $λ$.
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_θ,\mathfrak{p}_θ,\mathfrak{g}_θ,\mathfrak{r}_θ,\mathfrak{t}_θ$ at uncountable regular cardinals $θ$. Motivated by a theorem of Raghavan-Shelah who proved that $\mathfrak{s}_θ\leq\mathfrak{b}_θ$, we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak{s}_θ$ with $\mathfrak{p}_θ$ and $\mathfrak{g}_θ$. In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results concerning $\mathfrak{r}_θ$ from hyper-measurability assumptions, results which were previously known to be consistent only from supercompactness assumptions. In doing so, we answer several questions which appeared in the literature.
Let inv denote the cardinal invariants Depth^+ and Length^+ on Boolean algebras. For many singular cardinals we create a strict inequality between the product of the inv values and the inv of the product algebra. The proof holds in ZFC.
We prove that the consistency of the existence of a Dowker filter at $κ^+$ along with $2^κ=κ^+$ where $κ$ is regular and uncountable. Using Magidor forcing we also prove the consistency of the existence of a Dowker filter at $μ^+$ where $μ>{\rm cf}(μ)>ω$.
We prove that $α_M(λ)$ can be successor of a supercompact cardinal, when $λ$ is a Magidor cardinal. From this result we obtain the consistency of $α_M(λ)$ being a successor of a singular cardinal with uncountable cofinality.
We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.
We prove that consistently every bipartite graph of size $κ^+\timesκ^+$ contains either a clique or an independent subset of size $τ\timesτ$ for every $τ\inκ^+$, where $κ$ is a successor cardinal.
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 analyze the non-stationary ideal and the club filter at aleph_1 under MM.
We prove combinatorial theorems concerning the stick principle and cardinal characteristics.
We prove the consistency of the failure of the weak diamond $Φ_λ$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $Ψ_λ$ is equivalent to the statement $2^{<λ}<2^λ$ and hence holds at every strongly inaccessible cardinal.