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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 19 recordsLinked to original sources
arXiv2026

Representing a ring as the endomorphism ring of an abelian group

We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal $μ$ with $μ^+ < 2^μ< 2^{μ^+}$, and for a wide class of cotorsion-free rings, we construct $μ^+$-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of $κ$-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.

math.LO↗
arXiv2026

First-Order Laws for Random Geometric Graphs on the Torus

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.

math.PR↗
arXiv2026

Limit laws for component-pruned sparse random graphs and percolated tori

We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erdős-Rényi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfies a zero-one law for $\mathrm{MSO}_2$, with quantification over sets of vertices and sets of edges. The proof combines uniform component counts, an MSO Feferman-Vaught decomposition for disjoint unions, and semilinearity of the order spectra of MSO-definable classes of finite trees. We also show that the term $f(n)\log f(n)$ cannot simply be omitted: star components can occur at first-order-visible Poisson thresholds. We further establish first-order limit laws for bond percolation on the discrete torus $T_L^d$. In the two-sided subpolynomial regime, pruning below a sufficiently slow threshold yields a zero-one law. For the unpruned model in either one-sided polynomial regime, the reciprocal exponents $α=1/k$ are precisely the critical scales. At such a scale, an extended limit of $N p_N^k$ or $N q_N^k$ equal to $0$ or $\infty$ gives a zero-one law; a positive finite limit gives a convergence law but not a zero-one law; and the absence of an extended limit gives failure of convergence. Finally, $\mathrm{MSO}_1$ already detects the parity of the torus side length through bipartiteness, producing a natural obstruction to monadic convergence in a near-deterministic regime.

math.LO↗
arXiv2026

Iterated Ramsey bounds for the Hales-Jewett numbers

Consider the Hales-Jewett theorem. The $k$-dimensional version of it tells us that the combinatorial space $\mathcal{U}_{M, Λ} = \{ η\mid η: M \to Λ\}$ has, under suitable assumptions, monochromatic $k$-dimensional subspaces, where by a $k$-dimensional subspace we mean there exist a partition $\langle N_0, N_1, \cdots, N_k \rangle$ of $M$ such that $N_1, \cdots, N_k \neq \emptyset$ (but we allow $N_0$ to be empty) and some $ρ_0: N_0 \to Λ$, such that the subspace consists of those $ρ\in \mathcal{U}_{M, Λ}$ such that for $0<l<k+1, ρ\restriction N_l$ is constant and $ρ\restriction N_0= ρ_0.$ It seems natural to think it is better to have each $N_{l}, 0<l<k+1$ a singleton. However it is then impossible to always find monochromatic $k$-dimensional subspaces (for example color $η$ by $0$ if $|η^{-1}\{α\}|$ is an even number and by $1$ otherwise). But modulo restricting the sign of each $|η^{-1}\{α\}|$, we prove the parallel theorem -- whose proof is not related to the Hales-Jewett theorem. We then connect the two numbers by showing that the Hales-Jewett numbers are not too much above the present ones. This gives an alternative proof of the Hales-Jewett theorem.

math.CO↗
arXiv2026

HJ numbers revisited

We improve the bounds on the Hales-Jewett numbers to a tower of exponentiations. Earlier it was $WaW$ (that is, iterations of towers which are themselves iterated exponentiations). We improve the inductive step there (induction on the size of the alphabet, $|Λ|$) to 2-exponentiations, instead of towers. In the longer work in typing, (A) We present this inductive step as a partition theorem in its own right; (but in this preliminary version we make it just serve the bound on HJ numbers). (B) We shall deal with the density version of Hales-Jewett with similar bound. We are also dealing with the Graham-Rothschild Theorem and the Affine Ramsey Theorem and the polynomial case, and give background.

math.CO↗
arXiv2026

From Shelah's block-content to Hales-Jewett

We study the quantitative relationship between the Hales-Jewett numbers and Shelah's block-content canonization functions. Block-content canonization yields a block subspace on which the color of a word is determined solely by the multiplicities of the alphabet letters among the variable blocks. We show that this canonical information, combined with the multidimensional Gallai-Witt theorem, suffices to produce a monochromatic Hales-Jewett subspace. The argument passes to the space of content vectors, finds a monochromatic homothetic copy of a finite content simplex, and lifts it through the canonical block subspace. Combined with the elementary fact that Hales-Jewett bounds block canonization, this gives a two-way quantitative comparison up to an explicit change of parameters. The underlying mechanism may be summarized by the slogan Hales-Jewett = block canonization + Gallai-Witt We also prove the corresponding equal-block result and show, by an explicit coloring over every finite field of odd prime order, that the analogous singleton-coordinate canonization principle fails as soon as two coordinates remain live.

math.CO↗
arXiv2026

Discontinuous homomophisms without Hamel bases

We produce a model of ZF + DC in which there exists a discontinuous homomorphism from the real line to itself but no Hamel basis for the real line, and prove a generalization of this result in terms of internal direct sums.

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

Modules and Infinitary Logics

We prove that the theory of abelian groups and R-modules even in infinitary logic is stable and understood to some extent.

math.LO↗
arXiv2026

Model-theoretic Tameness in finite extensions of groups

It is shown that finite-index extensions and finite-index subgroups of $ω$-stable groups can be model-theoretically wild. More precisely, there exists an $ω$-stable group $G$ such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of $G$ and in some finite-index subgroup of $G$.

math.LO↗
arXiv2026

A complicated family of trees with omega + 1 levels

Our aim is to prove that if T is a complete first order theory, which is not superstable (no knowledge on this notion is required), included in a theory T_1 then for any lambda > |T_1| there are 2^lambda models of T_1 such that for any two of them the tau(T)-reducts of one is not elementarily embeddable into the tau(T)-reduct of the other, thus completing the investigation of [Sh:a, Ch. VIII]. Note the difference with the case of unstable T: there lambda > |T_1| + aleph_0 suffices. By [Sh:E59] it suffices for every such lambda to find a complicated enough family of trees with omega + 1 levels of cardinality lambda. If lambda is regular this is done already in [Sh:c, Ch. VIII]. The proof here (in sections 1,2) go by dividing to cases, each with its own combinatorics. In particular we have to use guessing clubs which was discovered for this aim. In S.3 we consider strongly aleph_varepsilon-saturated models of stable T (so if you do not know stability better just ignore this). We also deal with separable reduced Abelian p-groups. We then deal with various improvements of the earlier combinatorial results.

math.LO↗
arXiv2026

On the problem of stability of abstract elementary classes of modules

It is an open problem of Mazari-Armida whether every abstract elementary class of $R$-modules $(\mathbf{K}, \leq_{\mathrm{pure}})$, with $\leq_{\mathrm{pure}}$ the pure submodule relation, is stable. We answer this question in the negative by constructing unstable abstract elementary classes $(\mathbf{K}, \leq_{\mathrm{pure}})$ of torsion-free abelian groups. On the other hand, we prove (in $\mathrm{ZFC}$) that if $R$ is any ring and $(\mathbf{K}, \preccurlyeq)$ is an abstract elementary class of $R$-modules which is $κ$-local (also called $κ$-tame) for some $κ\geq \mathrm{LS}(\mathbf{K}, \preccurlyeq)$, then $(\mathbf{K}, \preccurlyeq)$ is almost stable, where almost stability is a new notion of independent interest that we introduce in this paper, and which is equivalent to the usual notion of stability under the assumption of amalgamation. As a consequence, assuming the existence of a strongly compact cardinal $κ$, we have that every abstract elementary class $(\mathbf{K}, \preccurlyeq)$ of $R$-modules with amalgamation satisfying $κ> \mathrm{LS}(\mathbf{K}, \preccurlyeq)$ is stable.

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↗
arXiv2026

$\aleph_1$-free abelian non-Archimedean Polish groups

An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^ω$; more strongly, $\mathbb{Z}^ω$ is separable. In this paper we investigate $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so $\aleph_1$-free) abelian non-Archimedean Polish groups which are pairwise not topologically isomorphic. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_\infty$: separability, torsionlessness, $\aleph_1$-freeness and $\mathbb{Z}$-homogeneity.

math.LO↗
arXiv2026

Homogeneous forcing

Assume $κ= κ^{< κ}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{κ>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this, which we will focus on, is for the forcings to have support $<κ$ and the $κ^+$-cc, and be strategically $<κ$-complete. The aim is to have homogeneous forcings, so that the iteration has many automorphisms. In addition to the inherent interest, such iterations are helpful for considering some natural ideals on ${}^\kappa2$, in order to get a model of ${\rm ZF} + {\rm DC}_κ +$ ``modulo this ideal, every set is equivalent to a $κ$-Borel one." But here we only have many automorphisms of the index set $L$ and therefore of the iteration of iterands $\mathbb{Q} $; we do not necessarily have homogeneity of $\mathbb{Q} $, and we do not have automorphisms mapping other names of $\mathbb{Q} $-reals onto each other. %\notemgrimes{What are the other names? Where do they come from?} However, for some reasonable forcing notions, there are no other $\mathbb{Q} $-reals! This was the reason for introducing and investigating saccharinity in earlier works with Jakob Kellner and with Haim Horowitz.

math.LO↗
arXiv2026

No universal group in a cardinal

For many classes of models, there are universal members in any cardinal $λ$ which "essentially satisfies GCH", i.e. $λ= 2^{< λ}$, in particular for the class of a complete first order $T$ (well, if at least $λ> |T|$). But if the class is "complicated enough", e.g. the class of linear orders, we know that if $λ$ is "regular and not so close to satisfying GCH" then there is no universal member. Here we find new sufficient conditions (which we call the olive property), not covered by earlier cases (i.e. fail the so-called SOP$_4$). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions.

math.LO↗
arXiv2026

A unique $Q$-point and infinitely many near-coherence classes of ultrafilters

We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-$ω_2$ countable support iteration of restricted Mathias forcing over a ground model satisfying $\textsf{CH}$, there is a unique $Q$-point up to isomorphism. In particular, it is consistent that there is only one $Q$-point while there are $2^{\mathfrak{c}}$-many near-coherence classes of ultrafilters.

math.LO↗