Adi Jarden | Ariel University (original) (raw)
Papers by Adi Jarden
We point out some connections between existence of homogenous sets for certain edge colorings and... more We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors, µ satisfies µ + < κ. Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property and for each λ, µ < κ there exists κ * < κ such that every tree of height µ with λ nodes has less than κ * branches.
We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: study... more We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking λ^+-frame from a semi-good non-forking λ-frame. But the classes K_λ^+ and ≼ K_λ^+ are replaced: K_λ^+ is restricted to the saturated models and the partial order ≼ K_λ^+ is restricted to the partial order ≼^NF_λ^+. Here, we avoid the restriction of the partial order ≼ K_λ^+, assuming that every saturated model (in λ^+ over λ) is an amalgamation base and (λ,λ^+)-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that M ≼ M^+ if and only if M ≼^NF_λ^+M^+, provided that M and M^+ are saturated models. We present sufficient conditions for three good non-forking λ^+-fr...
We present a connection between tameness and non-forking frames. In addition we improve results a... more We present a connection between tameness and non-forking frames. In addition we improve results about independence and dimension.
A relevant collection is a collection, F, of sets, such that each set in F has the same cardinali... more A relevant collection is a collection, F, of sets, such that each set in F has the same cardinality, α(F). A Konig Egervary (KE) collection is a relevant collection F, that satisfies | F|+| F|=2α(F). An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In jlm and dam, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In hke, Jarden characterize hke collections. Let Γ be a relevant collection such that Γ-{S} is an hke collection, for every S ∈Γ. We study the difference between |Γ_1-Γ_2| and |Γ_2-Γ_1|, where {Γ_1,Γ_2} is a partition of Γ. We get new characterizations for an hke collection and for a KE graph.
Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertice... more Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary (KE in short) graph if α(G) + μ(G)= |V(G)|, where α(G) denotes the size of a maximum independent set and μ(G) is the cardinality of a maximum matching. Let Ω(G) denote the family of all maximum independent sets. A collection F of sets is an hke collection if |Γ|+|Γ|=2α holds for every subcollection Γ of F. We characterize an hke collection and invoke new characterizations of a KE graph. We prove the existence and uniqueness of a graph G such that Ω(G) is a maximal hke collection. It is a bipartite graph. As a result, we solve a problem of Jarden, Levit and Mandrescu jlm, proving that F is an hke collection if and only if it is a subset of Ω(G) for some graph G and | F|+| F|=2α(F). Finally, we show that the maximal cardinality of an hke collection F with α(F)=α and | F|=n is 2^n-α.
Advances and Applications in Discrete Mathematics, 2019
arXiv: Combinatorics, 2016
A relevant collection is a collection, FFF, of sets, such that each set in FFF has the same cardi... more A relevant collection is a collection, FFF, of sets, such that each set in FFF has the same cardinality, alpha(F)\alpha(F)alpha(F). A Konig Egervary (KE) collection is a relevant collection FFF, that satisfies ∣bigcupF∣+∣bigcapF∣=2alpha(F)|\bigcup F|+|\bigcap F|=2\alpha(F)∣bigcupF∣+∣bigcapF∣=2alpha(F). An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In \cite{jlm} and \cite{dam}, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In \cite{hke}, Jarden characterize hke collections. Let Gamma\GammaGamma be a relevant collection such that Gamma−S\Gamma-\{S\}Gamma−S is an hke collection, for every SinGammaS \in \GammaSinGamma. We study the difference between ∣bigcapGamma1−bigcupGamma2∣|\bigcap \Gamma_1-\bigcup \Gamma_2|∣bigcapGamma1−bigcupGamma2∣ and ∣bigcapGamma2−bigcupGamma1∣|\bigcap \Gamma_2-\bigcup \Gamma_1|∣bigcapGamma2−bigcupGamma1∣, where Gamma1,Gamma2\{\Gamma_1,\Gamma_2\}Gamma_1,Gamma_2 is a partition of Gamma\GammaGamma. We get new characterizations for an hke collection and for a KE graph.
arXiv: Combinatorics, 2016
Let GGG be a simple graph with vertex set V(G)V(G)V(G). A subset SSS of V(G)V(G)V(G) is independent if no two... more Let GGG be a simple graph with vertex set V(G)V(G)V(G). A subset SSS of V(G)V(G)V(G) is independent if no two vertices from SSS are adjacent. The graph GGG is known to be a Konig-Egervary (KE in short) graph if alpha(G)+mu(G)=∣V(G)∣\alpha(G) + \mu(G)= |V(G)|alpha(G)+mu(G)=∣V(G)∣, where alpha(G)\alpha(G)alpha(G) denotes the size of a maximum independent set and mu(G)\mu(G)mu(G) is the cardinality of a maximum matching. Let Omega(G)\Omega(G)Omega(G) denote the family of all maximum independent sets. A collection FFF of sets is an hke collection if ∣bigcupGamma∣+∣bigcapGamma∣=2alpha|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha∣bigcupGamma∣+∣bigcapGamma∣=2alpha holds for every subcollection Gamma\GammaGamma of FFF. We characterize an hke collection and invoke new characterizations of a KE graph. We prove the existence and uniqueness of a graph GGG such that Omega(G)\Omega(G)Omega(G) is a maximal hke collection. It is a bipartite graph. As a result, we solve a problem of Jarden, Levit and Mandrescu \cite{jlm}, proving that FFF is an hke collection if and only if it is a subset of Omega(G)\Omega(G)Omega(G) for some graph GGG and ∣bigcupF∣+∣bigcapF∣=2alpha(F)|\bigcup F|+|\bigcap F|=2\alpha(F)∣bigcupF∣+∣bigcapF∣=2alpha(F). Finally, we ...
Advances and Applications in Discrete Mathematics, 2021
arXiv preprint arXiv:1001.2876, Jan 17, 2010
3 an independence relation and prove that one can define dimension by it. Sections 1,2,3 are self... more 3 an independence relation and prove that one can define dimension by it. Sections 1,2,3 are self contained. In section 4 assuming existence of uniqueness triples (so familiarity with [JrSh 875], [Sh 600] or [Sh 705] is assumed), We prove that the relations independence and finitely independence are the same and more properties. What are the connections between the present paper and other papers? In [JrSh 875] we study stability theory without assuming stability, but weak stability. The main purpose is to study abstract elementary classes (shortly aec's) ...
We work with a pre-λ-frame, which is an abstract elementary class (AEC) endowed with a collection... more We work with a pre-λ-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality λ. We investigate the density of uniqueness triples in a given pre-λ-frame s, that is, under what circumstances every basic triple admits a non-forking extension that is a uniqueness triple. Prior results in this direction required strong hypotheses on s. Our main result is an improvement, in that we assume far fewer hypotheses on s. In particular, we do not require s to satisfy the extension, uniqueness, stability, or symmetry properties, or any form of local character, though we do impose the amalgamation and stability properties in λ^+, and we do assume (λ^+). As a corollary, by applying our main result to the trivial λ-frame, it follows that in any AEC K satisfying modest hypotheses on K_λ and K_λ^+, the set of *-domination triples in K_λ is dense among the non-algeb...
arXiv: Logic, 2018
We work with a \emph{$\lambda$-frame}, which is an abstract elementary class endowed with a colle... more We work with a \emph{$\lambda$-frame}, which is an abstract elementary class endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality lambda\lambdalambda. We will show that assuming the diamond axiom diamondsuit(lambda+)\diamondsuit(\lambda^+)diamondsuit(lambda+), any basic type admits a non-forking extension that has a \emph{uniqueness triple}. Prior results of Shelah in this direction required some form of diamondsuit\diamondsuitdiamondsuit at two consecutive cardinals as well as a constraint on the number of models of size lambda++\lambda^{++}lambda++.
Annals of Pure and Applied Logic, 2013
We presents an independence relation on sets, one can define dimension by it, assuming that we ha... more We presents an independence relation on sets, one can define dimension by it, assuming that we have an abstract elementary class with a forking notion that satisfies the axioms of a good frame minus stability.
We combine two approaches to the study of classification theory of AECs: (1) that of Shelah: stud... more We combine two approaches to the study of classification theory of AECs: (1) that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and (2) that of Grossberg and VanDieren [8]: (studying non-splitting) assuming the amalgamation property and tameness. In [9] we derive a good non-forking λ +-frame from a semi-good nonforking λ-frame. But the classes K λ + and ↾ K λ + are replaced: K λ + is restricted to the saturated models and the partial order ↾ K λ + is restricted to the partial order NF λ +. Here, we avoid the restriction of the partial order ↾ K λ + , assuming that every saturated model (in λ + over λ) is an amalgamation base and (λ, λ +)-tameness for non-forking types over saturated models, (in addition to the hypotheses of [9]): Theorem 7.15 states that M M + if and only if M NF λ + M + , provided that M and M + are saturated models. We present sufficient conditions for three good non-forking λ +-frames: one relates to all the models of cardinality λ + and the two others relate to the saturated models only. By an 'unproven claim' of Shelah, if we can repeat this procedure ω times, namely, 'derive' good non-forking λ +n frame for each n < ω then the categoricity conjecture holds. In [18], Vasey applies Theorem 7.8, proving the categoricity conjecture under the above 'unproven claim' of Shelah. In [12], we apply Theorem 7.15, proving the existence of primeness triples.
We point out some connections between existence of homogenous sets for certain edge colorings and... more We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors, µ satisfies µ + < κ. Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property and for each λ, µ < κ there exists κ * < κ such that every tree of height µ with λ nodes has less than κ * branches.
We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: study... more We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking λ^+-frame from a semi-good non-forking λ-frame. But the classes K_λ^+ and ≼ K_λ^+ are replaced: K_λ^+ is restricted to the saturated models and the partial order ≼ K_λ^+ is restricted to the partial order ≼^NF_λ^+. Here, we avoid the restriction of the partial order ≼ K_λ^+, assuming that every saturated model (in λ^+ over λ) is an amalgamation base and (λ,λ^+)-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that M ≼ M^+ if and only if M ≼^NF_λ^+M^+, provided that M and M^+ are saturated models. We present sufficient conditions for three good non-forking λ^+-fr...
We present a connection between tameness and non-forking frames. In addition we improve results a... more We present a connection between tameness and non-forking frames. In addition we improve results about independence and dimension.
A relevant collection is a collection, F, of sets, such that each set in F has the same cardinali... more A relevant collection is a collection, F, of sets, such that each set in F has the same cardinality, α(F). A Konig Egervary (KE) collection is a relevant collection F, that satisfies | F|+| F|=2α(F). An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In jlm and dam, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In hke, Jarden characterize hke collections. Let Γ be a relevant collection such that Γ-{S} is an hke collection, for every S ∈Γ. We study the difference between |Γ_1-Γ_2| and |Γ_2-Γ_1|, where {Γ_1,Γ_2} is a partition of Γ. We get new characterizations for an hke collection and for a KE graph.
Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertice... more Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary (KE in short) graph if α(G) + μ(G)= |V(G)|, where α(G) denotes the size of a maximum independent set and μ(G) is the cardinality of a maximum matching. Let Ω(G) denote the family of all maximum independent sets. A collection F of sets is an hke collection if |Γ|+|Γ|=2α holds for every subcollection Γ of F. We characterize an hke collection and invoke new characterizations of a KE graph. We prove the existence and uniqueness of a graph G such that Ω(G) is a maximal hke collection. It is a bipartite graph. As a result, we solve a problem of Jarden, Levit and Mandrescu jlm, proving that F is an hke collection if and only if it is a subset of Ω(G) for some graph G and | F|+| F|=2α(F). Finally, we show that the maximal cardinality of an hke collection F with α(F)=α and | F|=n is 2^n-α.
Advances and Applications in Discrete Mathematics, 2019
arXiv: Combinatorics, 2016
A relevant collection is a collection, FFF, of sets, such that each set in FFF has the same cardi... more A relevant collection is a collection, FFF, of sets, such that each set in FFF has the same cardinality, alpha(F)\alpha(F)alpha(F). A Konig Egervary (KE) collection is a relevant collection FFF, that satisfies ∣bigcupF∣+∣bigcapF∣=2alpha(F)|\bigcup F|+|\bigcap F|=2\alpha(F)∣bigcupF∣+∣bigcapF∣=2alpha(F). An hke (hereditary KE) collection is a relevant collection such that all of his non-empty subsets are KE collections. In \cite{jlm} and \cite{dam}, Jarden, Levit and Mandrescu presented results concerning graphs, that give the motivation for the study of hke collections. In \cite{hke}, Jarden characterize hke collections. Let Gamma\GammaGamma be a relevant collection such that Gamma−S\Gamma-\{S\}Gamma−S is an hke collection, for every SinGammaS \in \GammaSinGamma. We study the difference between ∣bigcapGamma1−bigcupGamma2∣|\bigcap \Gamma_1-\bigcup \Gamma_2|∣bigcapGamma1−bigcupGamma2∣ and ∣bigcapGamma2−bigcupGamma1∣|\bigcap \Gamma_2-\bigcup \Gamma_1|∣bigcapGamma2−bigcupGamma1∣, where Gamma1,Gamma2\{\Gamma_1,\Gamma_2\}Gamma_1,Gamma_2 is a partition of Gamma\GammaGamma. We get new characterizations for an hke collection and for a KE graph.
arXiv: Combinatorics, 2016
Let GGG be a simple graph with vertex set V(G)V(G)V(G). A subset SSS of V(G)V(G)V(G) is independent if no two... more Let GGG be a simple graph with vertex set V(G)V(G)V(G). A subset SSS of V(G)V(G)V(G) is independent if no two vertices from SSS are adjacent. The graph GGG is known to be a Konig-Egervary (KE in short) graph if alpha(G)+mu(G)=∣V(G)∣\alpha(G) + \mu(G)= |V(G)|alpha(G)+mu(G)=∣V(G)∣, where alpha(G)\alpha(G)alpha(G) denotes the size of a maximum independent set and mu(G)\mu(G)mu(G) is the cardinality of a maximum matching. Let Omega(G)\Omega(G)Omega(G) denote the family of all maximum independent sets. A collection FFF of sets is an hke collection if ∣bigcupGamma∣+∣bigcapGamma∣=2alpha|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha∣bigcupGamma∣+∣bigcapGamma∣=2alpha holds for every subcollection Gamma\GammaGamma of FFF. We characterize an hke collection and invoke new characterizations of a KE graph. We prove the existence and uniqueness of a graph GGG such that Omega(G)\Omega(G)Omega(G) is a maximal hke collection. It is a bipartite graph. As a result, we solve a problem of Jarden, Levit and Mandrescu \cite{jlm}, proving that FFF is an hke collection if and only if it is a subset of Omega(G)\Omega(G)Omega(G) for some graph GGG and ∣bigcupF∣+∣bigcapF∣=2alpha(F)|\bigcup F|+|\bigcap F|=2\alpha(F)∣bigcupF∣+∣bigcapF∣=2alpha(F). Finally, we ...
Advances and Applications in Discrete Mathematics, 2021
arXiv preprint arXiv:1001.2876, Jan 17, 2010
3 an independence relation and prove that one can define dimension by it. Sections 1,2,3 are self... more 3 an independence relation and prove that one can define dimension by it. Sections 1,2,3 are self contained. In section 4 assuming existence of uniqueness triples (so familiarity with [JrSh 875], [Sh 600] or [Sh 705] is assumed), We prove that the relations independence and finitely independence are the same and more properties. What are the connections between the present paper and other papers? In [JrSh 875] we study stability theory without assuming stability, but weak stability. The main purpose is to study abstract elementary classes (shortly aec's) ...
We work with a pre-λ-frame, which is an abstract elementary class (AEC) endowed with a collection... more We work with a pre-λ-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality λ. We investigate the density of uniqueness triples in a given pre-λ-frame s, that is, under what circumstances every basic triple admits a non-forking extension that is a uniqueness triple. Prior results in this direction required strong hypotheses on s. Our main result is an improvement, in that we assume far fewer hypotheses on s. In particular, we do not require s to satisfy the extension, uniqueness, stability, or symmetry properties, or any form of local character, though we do impose the amalgamation and stability properties in λ^+, and we do assume (λ^+). As a corollary, by applying our main result to the trivial λ-frame, it follows that in any AEC K satisfying modest hypotheses on K_λ and K_λ^+, the set of *-domination triples in K_λ is dense among the non-algeb...
arXiv: Logic, 2018
We work with a \emph{$\lambda$-frame}, which is an abstract elementary class endowed with a colle... more We work with a \emph{$\lambda$-frame}, which is an abstract elementary class endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality lambda\lambdalambda. We will show that assuming the diamond axiom diamondsuit(lambda+)\diamondsuit(\lambda^+)diamondsuit(lambda+), any basic type admits a non-forking extension that has a \emph{uniqueness triple}. Prior results of Shelah in this direction required some form of diamondsuit\diamondsuitdiamondsuit at two consecutive cardinals as well as a constraint on the number of models of size lambda++\lambda^{++}lambda++.
Annals of Pure and Applied Logic, 2013
We presents an independence relation on sets, one can define dimension by it, assuming that we ha... more We presents an independence relation on sets, one can define dimension by it, assuming that we have an abstract elementary class with a forking notion that satisfies the axioms of a good frame minus stability.
We combine two approaches to the study of classification theory of AECs: (1) that of Shelah: stud... more We combine two approaches to the study of classification theory of AECs: (1) that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and (2) that of Grossberg and VanDieren [8]: (studying non-splitting) assuming the amalgamation property and tameness. In [9] we derive a good non-forking λ +-frame from a semi-good nonforking λ-frame. But the classes K λ + and ↾ K λ + are replaced: K λ + is restricted to the saturated models and the partial order ↾ K λ + is restricted to the partial order NF λ +. Here, we avoid the restriction of the partial order ↾ K λ + , assuming that every saturated model (in λ + over λ) is an amalgamation base and (λ, λ +)-tameness for non-forking types over saturated models, (in addition to the hypotheses of [9]): Theorem 7.15 states that M M + if and only if M NF λ + M + , provided that M and M + are saturated models. We present sufficient conditions for three good non-forking λ +-frames: one relates to all the models of cardinality λ + and the two others relate to the saturated models only. By an 'unproven claim' of Shelah, if we can repeat this procedure ω times, namely, 'derive' good non-forking λ +n frame for each n < ω then the categoricity conjecture holds. In [18], Vasey applies Theorem 7.8, proving the categoricity conjecture under the above 'unproven claim' of Shelah. In [12], we apply Theorem 7.15, proving the existence of primeness triples.
קובץ הודעות משתנה מאפשר לי לכתב לכם הודעה אפילו מבלי להתחבר לאינטרנט. כאשר אכתב בו הודעה, אצטרך ל... more קובץ הודעות משתנה מאפשר לי לכתב לכם הודעה אפילו מבלי להתחבר לאינטרנט. כאשר אכתב בו הודעה, אצטרך להודיע לכם בהרצאה על כך.
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