Itay Kaplan
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View article: EXISTENTIALLY CLOSED MODELS OF FIELDS WITH A DISTINGUISHED SUBMODULE
EXISTENTIALLY CLOSED MODELS OF FIELDS WITH A DISTINGUISHED SUBMODULE Open
This article deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author.…
View article: Infinite cliques in simple and stable graphs
Infinite cliques in simple and stable graphs Open
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 st…
View article: High-Rate Nested-Lattice Quantized Matrix Multiplication with Small Lookup Tables
High-Rate Nested-Lattice Quantized Matrix Multiplication with Small Lookup Tables Open
Recent work have shown that the quantization for matrix multiplication problem can be optimally solved by quantizing each column in each matrix using a nested lattice code, and then multiplying the de-quantized matrices. It was further dem…
View article: Stable reducts of elementary extensions of Presburger arithmetic
Stable reducts of elementary extensions of Presburger arithmetic Open
Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < ω$, $[G: nG]$ is finite). Then every stable reduct of $N$ which expands $(G,+)$ (equivalently every reduct t…
View article: Density of compressible types and some consequences
Density of compressible types and some consequences Open
We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we o…
View article: Generic Stability Independence and Treeless Theories
Generic Stability Independence and Treeless Theories Open
We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both bina…
View article: ON LARGE EXTERNALLY DEFINABLE SETS IN NIP
ON LARGE EXTERNALLY DEFINABLE SETS IN NIP Open
We study cofinal systems of finite subsets of $\omega _1$ . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theor…
View article: Automorphisms of the Rado meet-tree
Automorphisms of the Rado meet-tree Open
We prove that the group of automorphisms of the generic meet-tree expansion of an infinite non-unary free Fra\"ıssé limit over a finite relational language is simple. As a prototypical case, the group of automorphism of the Rado meet-tree …
View article: Infinite stable graphs with large chromatic number II
Infinite stable graphs with large chromatic number II Open
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 \textup{Sh}_n(\omega) and thus has ele…
View article: Generic Stability Independence and Treeless theories
Generic Stability Independence and Treeless theories Open
We initiate a systematic study of \emph{generic stability independence} and introduce the class of \emph{treeless theories} in which this notion of independence is particularly well-behaved. We show that the class of treeless theories cont…
View article: The model theory of geometric random graphs
The model theory of geometric random graphs Open
We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, where two vertices are adjacent with probability $p>0…
View article: SATURATED MODELS FOR THE WORKING MODEL THEORIST
SATURATED MODELS FOR THE WORKING MODEL THEORIST Open
We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set…
View article: A definable $(p,q)$-theorem for NIP theories
A definable $(p,q)$-theorem for NIP theories Open
We prove a definable version of Matoušek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a proof of Boxall and Kestner who proved this theorem in the d…
View article: On large externally definable sets in NIP
On large externally definable sets in NIP Open
We study cofinal systems of finite subsets of $ω_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory, any …
View article: BOOLEAN TYPES IN DEPENDENT THEORIES
BOOLEAN TYPES IN DEPENDENT THEORIES Open
The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal …
View article: Existentially closed models of fields with a distinguished submodule
Existentially closed models of fields with a distinguished submodule Open
This paper deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author. H…
View article: On algebraically closed fields with a distinguished subfield
On algebraically closed fields with a distinguished subfield Open
This paper is concerned with the model-theoretic study of pairs $(K,F)$ where $K$ is an algebraically closed field and $F$ is a distinguished subfield of $K$ allowing extra structure. We study the basic model-theoretic properties of those …
View article: Density of compressible types and some consequences
Density of compressible types and some consequences Open
We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we o…
View article: Infinite Stable Graphs With Large Chromatic Number II
Infinite Stable Graphs With Large Chromatic Number II Open
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 elementar…
View article: ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE
ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE Open
We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.
View article: On uniform definability of types over finite sets for NIP formulas
On uniform definability of types over finite sets for NIP formulas Open
Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.
View article: Exact saturation in pseudo-elementary classes for simple and stable theories
Exact saturation in pseudo-elementary classes for simple and stable theories Open
We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple…
View article: On Kim-independence
On Kim-independence Open
We study NSOP _1 theories. We define Kim-independence , which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim’s lemma, loca…
View article: A generalisation of von Staudt’s theorem on cross-ratios
A generalisation of von Staudt’s theorem on cross-ratios Open
A generalisation 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 proje…
View article: Automorphism groups of finite topological rank
Automorphism groups of finite topological rank Open
We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group …
View article: Local character of Kim-independence
Local character of Kim-independence Open
We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of ele…
View article: A Generalization of von Staudt's Theorem on Cross-Ratios
A Generalization of von Staudt's Theorem on Cross-Ratios Open
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 proje…