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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 55 records · Page 3Linked to original sources
arXiv2025

Adding the constant evasion and constant prediction numbers to Cichoń's maximum

Let $\mathfrak{e}^\mathsf{const}_2$ be the constant evasion number, that is, the size of the least family $F\subseteq{}^ω2$ of reals such that for each predictor $π\colon {}^{<ω}2\to 2$ there is $x\in F$ which is not constantly predicted by $π$; and let $\mathfrak{v}_2^\mathsf{const}$ be the constant prediction number, that is, the size of the least family $Π_2$ of functions $π\colon {}^{<ω}2\to 2$ such that for each $x\in{}^ω2$ there is $π\inΠ_2$ that predicts constantly $x$. In this work, we show that the constant evasion number $\mathfrak{e}_2^{\mathrm{cons}}$ and the constant prediction number $\mathfrak{v}_2^\mathsf{const}$ can be added to Cichoń's maximum with distinct values.

math.LO↗
arXiv2025

Forcing Diamond and Applications to Iterability

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $κ= κ^{<κ} \geq\aleph_1$ implies that $κ$-supported iterations of $κ$-Sacks forcing do not collapse $κ^+$ and are $κ$-proper in the affirmative. The results pertain to other higher tree forcings.

math.LO↗
arXiv2025

A permutation group acting transitively on certain collections of models

It is shown, from $σ$-centered Martin's Axiom, that there exists a proper dense subgroup of the symmetric group on a countably infinite set whose natural action on sufficiently flexible relational structures is transitive. This allows us to give consistent positive answers to some questions of Peter M. Neumann from the 1980s.

math.GR↗
arXiv2025

An analogue of U-rank for atomic classes

For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.

math.LO↗
arXiv2025

More separations of cardinal characteristics of the strong measure zero ideal

Let $\mathcal{N}$ be the $σ$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open questions from the work of Brendle, the first author, and Mejía~\cite{BCM2}.

math.LO↗
arXiv2024

Expressive Power of Infinitary Logic and Absolute co-Hopfianity

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We use the infinitary logic to show that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.

math.LO↗
arXiv2024

Artinian groups of large cardinality

A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We show that this problem is essentially the same as an analogous question, regarding universal algebras, asked by Jónsson in the 1960s. We further show that these problems are the same as the so-called free subset problem. As a result, one can have a consistent strong negative answer (from a large cardinal assumption) as well as a consistent positive answer.

math.GR↗
arXiv2024

Abstract Corrected Iterations

We consider $(<λ)$-support iterations of a version of $(<λ)$-strategically complete $λ^+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by Judah and Shelah for finite support iterations of Suslin ccc forcing, namely that if $(\mathbb{P}_α, \underset{\sim}{\mathbb Q_β} : α\leq δ, β<δ)$ is a FS iteration of Suslin ccc forcing and $U\subseteq δ$ is sufficiently closed, then letting $\mathbb{P}_U$ be the iteration along $U$, we have $\mathbb{P}_U \lessdot \mathbb{P}_δ$.

math.LO↗
arXiv2024

$\leq_{SP}$ Can Have Infinitely Many Classes

Building off of recent results on Keisler's order, we show that consistently, $\leq_{SP}$ has infinitely many classes. In particular, we define the property of $\leq k$-type amalgamation for simple theories, for each $2 \leq k < ω$. If we let $T_{n, k}$ be the theory of the random $k$-ary, $n$-clique free random hyper-graph, then $T_{n, k}$ has $\leq k-1$-type amalgamation but not $\leq k$-type amalgamation. We show that consistently, if $T$ has $\leq k$-type amalgamation then $T_{k+1, k} \not \leq_{SP} T$, thus producing infinitely many $\leq_{SP}$-classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal $\leq_{SP}$-class is exactly the class of unsimple theories. Finally, we show that consistently, if $T$ has $<\aleph_0$-type amalgamation, then $T \leq_{SP} T_{rg}$, the theory of the random graph.

math.LO↗
arXiv2024

Infinite Cliques in Simple and Stable Graphs

Suppose that $G$ is a graph of cardinality $μ^+$ with chromatic number $χ(G)\geq μ^+$. One possible reason that this could happen is if $G$ contains a clique of size $μ^+$. We prove that this is indeed the case when the edge relation is stable. When $G$ is a random graph (which is simple but not stable), this is not true. But still if in general the complete theory of $G$ is simple, $G$ must contain finite cliques of unbounded sizes.

math.LO↗
arXiv2024

On groups well represented as automorphism groups of groups

Assuming Gödel's axiom of constructibility $\bold V=\bold L,$ we present a characterization of those groups $L$ for which there exist arbitrarily large groups $H$ such that $aut(H) \cong L$. In particular, we show that it suffices to have one such group $H$ such that the size of its center is bigger than $ 2^{|L |+\aleph_0}$.

math.GR↗
arXiv2024

On automorphisms of $\mathcal P(λ)/[λ]^{<λ}$

We investigate the statement ``all automorphisms of $\mathcal P(λ)/[λ]^{<λ}$ are trivial''. We show that MA implies the statement for regular uncountable $λ<2^{\aleph_0}$; that the statement is false for measurable $λ$ if $2^λ=λ^+$; and that for ``densely trivial'' it can be forced (together with $2^λ=λ^{++}$) for inaccessible $λ$.

math.LO↗
arXiv2024

Preservation of splitting families and cardinal characteristics of the continuum

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń's diagram, $\mathfrak{m}(2\text{-Knaster})$, $\mathfrak{p}$, $\mathfrak{h}$, the splitting number $\mathfrak{s}$ and the reaping number $\mathfrak{r}$.

math.LO↗
arXiv2024

Diamond on Kurepa trees

We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin's axiom. We then prove that this principle is nontrivial by showing it may consistently fail.

math.LO↗
arXiv2024

Categoricity and amalgamation for AEC and $ κ$ measurable

In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{κ, \aleph_{0}}$ is categorical in a cardinal $λ> κ$, and $κ$ is a measurable cardinal. There we prove that the class of model of $T$ of cardinality $<λ$ (but $\geq |T|+κ$) has the amalgamation property; this is a step toward understanding the character of such classes of models. In this revised version we replaced the class of models of $T$ by $\mathfrak k$, an AEC (abstract elementary class) which has LS-number ${<} \, κ,$ or at least which behave nicely for ultrapowers by $D$, a normal ultra-filter on $κ$. Presently sub-section \S1A deals with $T \subseteq \mathbb L_{κ^{+}, \aleph_{0}}$ (and so does a large part of the introduction and little in the rest of \S1), but otherwise, all is done in the context of AEC.

math.LO↗
arXiv2024

Strong partition relations below the power set: consistency, was Sierpinski right, II?

We continue here [She88] but we do not rely on it. The motivation was a conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2-> [omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section 5 we disprove this and give similar negative results. In section 3 we prove the consistency of the conjecture replacing omega_2 by 2^omega, which is quite large, starting with an Erdős cardinal. In section 1 we present iteration lemmas which are needed when we replace omega by a larger lambda and in section 4 we generalize a theorem of Halpern and Lauchli replacing omega by a larger lambda .

math.LO↗