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Yatir Halevi

Publications and source records attributed to Yatir Halevi.

28 records · Page 2Linked to original sources
arXiv2021

Infinite Stable Graphs With Large Chromatic Number II

We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(ω)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.

math.LO↗
arXiv2021

Infinite Stable Graphs With Large Chromatic Number

We prove that if $G=(V,E)$ is an $ω$-stable (respectively, superstable) graph with $χ(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(ω)$ for some $n$. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if $G$ is $ω$-stable with $\mathrm{U}(G)\leq 2$ we prove that $n\leq 2$ suffices.

math.LO↗
arXiv2020

Dp-minimal integral domains

It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.

math.LO↗
arXiv2019

Definable V-topologies, Henselianity and NIP

We initiate the study of definable V-topolgies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if $(K,v_1,v_2)$ is a bi-valued NIP field with $v_1$ henselian (resp. t-henselian) then $v_1$ and $v_2$ are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.

math.LO↗
arXiv2019

The dp-rank of abelian groups

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik-Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups $A$ such that there is only finitely many primes $p$ such that the group $A/pA$ is infinite and for every prime $p$, there is only finitely many natural numbers $n$ such that $(p^nA)[p]/(p^{n+1}A)[p]$ is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.

math.LO↗
arXiv2018

A Conjectural Classification of Strongly Dependent Fields

We survey the history of Shelah's conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.

math.LO↗
arXiv2018

Eliminating Field Quantifiers in Strongly Dependent Henselian Fields

We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent then so is its henselization.

math.LO↗
arXiv2018

A Generalization of von Staudt's Theorem on Cross-Ratios

A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

math.AG↗
arXiv2016

Semigroups in Stable Structures

Assume $G$ is a definable group in a stable structure $M$. Newelski showed that the semigroup $S_G(M)$ of complete types concentrated on $G$ is an inverse limit of the $\infty$-definable (in $M^{eq}$) semigroups $S_{G,Δ}(M)$. He also shows that it is strongly $π$-regular: for every $p\in S_{G,Δ}(M)$ there exists $n\in\mathbb{N}$ such that $p^n$ is in a subgroup of $S_{G,Δ}(M)$. We show that $S_{G,Δ}(M)$ is in fact an intersection of definable semigroups, so $S_G(M)$ is an inverse limit of definable semigroups and that the latter property is enjoyed by all $\infty$-definable semigroups in stable structures.

math.LO↗