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Shimon Garti

Publications and source records attributed to Shimon Garti.

At least 19 recordsLinked to original sources
arXiv2026

Quadruples and cubes

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

math.LO↗
arXiv2026

Splendid extensions

Let $\kappa$ be a successor cardinal. We force a universe in which every model of PA of size $\kappa$ extends to a model M of the same size, where M has no splendid extensions. If there is an ineffable cardinal then this statement holds at some cardinal below it, in ZFC.

math.LO↗
arXiv2025

Echeloned saturation and forcing axioms

Addressing a question of Paul Larson we prove the following statement. If Chang's conjecture fails, Martin's axiom holds and the continuum is greater than $\aleph_2$, there are no weakly Laver ideals over $\aleph_1$. We also prove that under Baumgartner's axiom there are no Laver ideals over $\aleph_2$.

math.LO↗
arXiv2025

Jonsson and Magidor filters

We study the filter versions of square bracket partition relations, focusing on Jonssonicity and Magidority. We show that the singular cardinals in a Kleinberg sequence above some strong partition cardinal are not Magidor, but the limit of the sequence is Magidor. This is done under AD. We also force over a model of AD to obtain a singular cardinal carrying a Magidor filter.

math.LO↗
arXiv2025

On a problem of Erdos and Hajnal

We address a question of Erd\H{o}s and Hajnal about the ordinary partition relation $\aleph_{\omega+1}\nrightarrow(\aleph_{\omega+1},(3)_{\aleph_0})^2$. For $\theta=\mathrm{cf}(\lambda)<\lambda$, assuming $2^\lambda=\lambda^+$ they proved the negative relation $\lambda^+\nrightarrow(\lambda^+,(3)_\theta)^2$ and asked whether the (local instance of) GCH is indispensable. We show that this negative relation is consistent with $\lambda$ being a strong limit and $2^{\lambda}>\lambda^+$. The result can be pushed down to $\aleph_{\omega}$.

math.LO↗
arXiv2024

Hungarian Cubes

We prove a positive polarized cube relation for infinite cardinals.

math.LO↗
arXiv2024

Weak diamond and pcf theory

We obtain bounds on the cardinality of $pcf(\mathfrak{a})$ from instances of weak diamond. Consequently, under mild assumptions there are many singular cardinals of the from $\aleph_δ$ for which $2^{\aleph_δ}<\aleph_{(|δ|^{+3})}$. For example, if every limit cardinal is a strong limit cardinal then this bound holds at a class of singular cardinals.

math.LO↗
arXiv2023

An almost strong relation

Let $μ$ be a strong limit singular cardinal. We prove that if $2^μ > μ^+$ then $\binom{μ^+}μ\to \binomτμ_{<{\rm cf}(μ)}$ for every ordinal $τ<μ^+$. We obtain an optimal positive relation under $2^μ= μ^+$, as after collapsing $2^μ$ to $μ^+$ this positive relation is preserved.

math.LO↗
arXiv2023

Tiltan and superclub

We force superclub with an arbitrary large value of cov($\mathscr{M}$). We force tiltan with an arbitrary large value of add($\mathscr{M}$). Finally, we obtain a negative square bracket relation from superclub.

math.LO↗
arXiv2023

Superclub, splitting, separating statements

We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_κ=κ^+$. Based on this statement, we separate tiltan from superclub at a successor of a supercompact cardinal. We use Galvin's property in order to separate tiltan from superclub at successors of both regular and singular cardinals.

math.LO↗
arXiv2023

Tiltan

We prove that tiltan is consistent with the negation of Galvin's property. On the other hand, superclub implies Galvin's property. We also show that tiltan is consistent with a large value of the splitting number at kappa, where kappa is supercompact.

math.LO↗
arXiv2022

Non-Galvin Filters

We address the question of the consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions \cite[Questions 7.8,7.9]{TomMotiII}, \cite[Question 5]{NegGalSing} and improve theorem \cite[Theorem 2.3]{NegGalSing}.

math.LO↗
arXiv2022

Galvin's property at large cardinals and an application to partition calculus

In the first part of this paper, we explore the possibility for a very large cardinal $κ$ to carry a $κ$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $κ$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $κ$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $λ\rightarrow(λ,ω+1)^2$.

math.LO↗
arXiv2021

Negating the Galvin Property

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

math.LO↗