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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 $λ$-terraced cube relation fails whenever $λ$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $λ$. If $λ$ 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 $κ$ be a successor cardinal. We force a universe in which every model of PA of size $κ$ 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↗
arXiv2026

On a problem of Erdos and Hajnal

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

math.LO↗
arXiv2026

Length and ultraproducts

We construct, in ZFC, a sequence of Boolean algebras for which the product of Lengths is strictly smaller than the Length of the product algebra.

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↗
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↗