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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 199 records · Page 11Linked to original sources
arXiv2019

LF groups, aec amalgamation, few automorphisms

In S. 1 we deal with amalgamation bases, e.g., we define when an a.e.c. $k$ has $(λ,κ)$-amalgamation which means "many" M in $K^k_λ$ are amalgamation bases. We then consider what happens for the class of lf groups. In S. 2 we deal with weak definability of $a \in N\setminus M$ over $M$, for $ K_{exlf}$. In S. 3 we deal with indecomposable members of $K_{exlf}$ and with the existence of universal members of $K^k_μ$, for $μ$ strong limit of cofinality $\aleph_0$. Most noteworthy: if $K_{lf}$ has a universal model in $μ$ then it has a canonical one similar to the special models, (the parallel to saturated ones in this cardinality). In S. 4 we prove "every $G \in K^{lf}_<λ$ can be extended to a complete $(λ,θ)$-full G" for many cardinals. In a continuation we may consider "all the cardinals" or at least "almost all the cardinals"; also, we may consider a priori fixing the outer automorphism group.

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arXiv2019

PCF without choice

We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can be generalized. E.g., there is a class of regular cardinals, and we can prove cardinal inequality.

math.LO↗
arXiv2019

Infinite monochromatic sumsets for colourings of the reals

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb R$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb R\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb R$ so that $c$ is constant on $X+X$.

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arXiv2018

On finitary Hindman's numbers

Spencer asked whether the Paris-Harrington version of the Folkman-Sanders theorem has primitive recursive upper bounds. We give a positive answer to this question.

math.CO↗
arXiv2018

Ramsey theory for highly connected monochromatic subgraphs

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.

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arXiv2018

Cardinal characteristics at aleph omega

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 $o(μ)=μ^{++}$.

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arXiv2018

Automorphism Groups of Countable Stable Structures

For every countable structure $M$ we construct an $\aleph_0$-stable countable structure $N$ such that $Aut(M)$ and $Aut(N)$ are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure $M$ from the topological properties of the Polish group $Aut(M)$.

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arXiv2018

On the definability of mad families of vector spaces

We consider the definability of mad families in vector spaces of the form $\underset{n<ω}{\bigoplus} F$ where $F$ is a field of cardinality $\leq \aleph_0$. We show that there is no analytic mad family of subspaces when $F=\mathbb{F}_2$, partially answering a question of Smythe. Our proof relies on a variant of Mathias forcing restricted to a certain idempotent ultrafilter whose existence follows from Glazer's proof of Hindman's theorem.

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arXiv2018

The Strong Small Index Property for Free Homogeneous Structures

We show that in algebraically locally finite countable homogeneous structures with a free stationary independence relation the small index property implies the strong small index property. We use this and the main result of [15] to deduce that countable free homogeneous structures in a locally finite relational language have the strong small index property. We also exhibit new continuum sized classes of $\aleph_0$-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.

math.LO↗
arXiv2018

Another ordering of the ten cardinal characteristics in Cichoń's diagram

It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) <\mathrm{add}(\mathrm{Meager})=\mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) < \mathrm{non}(\mathrm{Null}) < \mathrm{cof}(\mathrm{Meager})= \mathfrak{d} < \mathrm{cof}(\mathrm{Null}) < 2^{\aleph_0}. \]

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arXiv2018

Polish Topologies for Graph Products of Groups

We give strong necessary conditions on the admissibility of a Polish group topology for an arbitrary graph product of groups $G(Γ, G_a)$, and use them to give a characterization modulo a finite set of nodes. As a corollary, we give a complete characterization in case all the factor groups $G_a$ are countable.

math.LO↗
arXiv2018

Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

Let $\mathbf{K}$ be the class of countable structures $M$ with the strong small index property and locally finite algebraicity, and $\mathbf{K}_*$ the class of $M \in \mathbf{K}$ such that $acl_M(\{ a \}) = \{ a \}$ for every $a \in M$. For homogeneous $M \in \mathbf{K}$, we introduce what we call the expanded group of automorphisms of $M$, and show that it is second-order definable in $Aut(M)$. We use this to prove that for $M, N \in \mathbf{K}_*$, $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $(Aut(M), M)$ and $(Aut(N), N)$ are isomorphic as permutation groups. In particular, we deduce that for $\aleph_0$-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's well-known $\forall \exists$-interpretation technique of [7]. Finally, we show that every finite group can be realized as the outer automorphism group of $Aut(M)$ for some countable $\aleph_0$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity.

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arXiv2018

The Higher Cichoń Diagram

For a strongly inacessible cardinal $κ$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}κ$-box product topology - the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799) - the ideal of nowhere stationary subsets of a (naturally defined) stationary set $S_{\rm pr}^κ\subseteq κ$. In particular, we analyse the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case ($κ=ω$) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null)$\le$non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with $\mathord<κ$-support (and a strong "Knaster" version of $κ^+$-cc) and one iteration with ${\le}κ$-support (and a version of $κ$-properness).

math.LO↗
arXiv2018

Dear Lambda

We prove the consistency of $\mathfrak{r}_λ<\mathfrak{d}_λ$ and even $\mathfrak{u}_λ<\mathfrak{d}_λ$ for a singular cardinal $λ$.

math.LO↗
arXiv2018

$κ$-Madness and Definability

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $κ$, there are no $Σ^1_1(κ)-κ-$mad families.

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