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Adi Jarden

Publications and source records attributed to Adi Jarden.

14 recordsLinked to original sources
arXiv2026

Combinatorial covering properties in an uncountable setting: canonical examples

We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and $\gamma$-properties in an uncountable setting. Our approach is motivated by canonical constructions from the classical countable case, including the examples of Bartoszy\'nski and Shelah separating the Hurewicz property from $\sigma$-compactness, the examples of Tsaban and Zdomskyy separating the Hurewicz and Menger properties, and Tsaban's construction of a nontrivial set of reals with the $\gamma$-property. We focus on the genuinely nontrivial aspects of these higher-cardinal generalizations, uncovering several open problems whose nature appears substantially different from that of their countable counterparts.

math.GN↗
arXiv2022

A Note on Edge Colorings and Trees

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.

math.LO↗
arXiv2018

Density of uniqueness triples from the diamond axiom

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 $\mathfrak s$, 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 $\mathfrak s$. Our main result is an improvement, in that we assume far fewer hypotheses on $\mathfrak s$. In particular, we do not require $\mathfrak s$ 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 $\diamondsuit(λ^+)$. As a corollary, by applying our main result to the trivial $λ$-frame, it follows that in any AEC $\mathbf K$ satisfying modest hypotheses on $\mathbf K_λ$ and $\mathbf K_{λ^+}$, the set of $*$-domination triples in $\mathbf K_λ$ is dense among the non-algebraic triples. We also apply our main result to the non-splitting relation, obtaining the density of uniqueness triples from very few hypotheses.

math.LO↗
arXiv2017

Duality and Hereditary König-Egerváry Set-systems

A König-Egerváry graph is a graph $G$ satisfying $α(G)+μ(G)=|V(G)|$, where $α(G)$ is the cardinality of a maximum independent set and $μ(G)$ is the matching number of $G$. Such graphs are those that admit a matching between $V(G)-\bigcup Γ$ and $\bigcap Γ$ where $Γ$ is a set-system comprised of maximum independent sets satisfying $|\bigcup Γ'|+|\bigcap Γ'|=2α(G)$ for every set-system $Γ' \subseteq Γ$; in order to improve this characterization of a König-Egerváry graph, we characterize \emph{hereditary König-Egerváry set-systems} (HKE set-systems, here after). An \emph{HKE} set-system is a set-system, $F$, such that for some positive integer, $α$, the equality $|\bigcup Γ|+|\bigcap Γ|=2α$ holds for every non-empty subset, $Γ$, of $F$. We prove the following theorem: Let $F$ be a set-system. $F$ is an HKE set-system if and only if the equality $|\bigcap Γ_1-\bigcup Γ_2|=|\bigcap Γ_2-\bigcup Γ_1|$ holds for every two non-empty disjoint subsets, $Γ_1,Γ_2$ of $F$. This theorem is applied in \cite{hke},\cite{broken}.

math.CO↗
arXiv2016

Hereditary Konig Egervary Collections

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 $|\bigcup Γ|+|\bigcap Γ|=2α$ 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 \cite{jlm}, proving that $F$ is an hke collection if and only if it is a subset of $Ω(G)$ for some graph $G$ and $|\bigcup F|+|\bigcap F|=2α(F)$. Finally, we show that the maximal cardinality of an hke collection $F$ with $α(F)=α$ and $|\bigcup F|=n$ is $2^{n-α}$.

math.CO↗
arXiv2016

The First Time KE is Broken up

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 $|\bigcup F|+|\bigcap F|=2α(F)$. 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 $Γ$ be a relevant collection such that $Γ-\{S\}$ is an hke collection, for every $S \in Γ$. We study the difference between $|\bigcap Γ_1-\bigcup Γ_2|$ and $|\bigcap Γ_2-\bigcup Γ_1|$, where $\{Γ_1,Γ_2\}$ is a partition of $Γ$. We get new characterizations for an hke collection and for a KE graph.

math.CO↗
arXiv2015

Tameness, Uniqueness and amalgamation

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 $\preceq \restriction K_{λ^+}$ are replaced: $K_{λ^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{λ^+}$ is restricted to the partial order $\preceq^{NF}_{λ^+}$. Here, we avoid the restriction of the partial order $\preceq \restriction K_{λ^+}$, 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 $M \preceq M^+$ if and only if $M \preceq^{NF}_{λ^+}M^+$, 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. Vasey applies one of our main theorems in a proof of the categoricity conjecture under the above `unproven claim' of Shelah and more assumptions. In [Jrprime], we apply the main theorem in a proof of the existence of primeness triples.

math.LO↗
arXiv2015

Monotonic Properties of Collections of Maximum Independent Sets of a Graph

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 if alpha(G) + mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. Let Omega(G) denote the family of all maximum independent sets, and f be the function from the set of subcollections Gamma of Omega(G) such that f(Gamma) = (the cardinality of the union of elements of Gamma) + (the cardinality of the intersection of elements of Gamma). Our main finding claims that f is "<<"-increasing, where the preorder {Gamma1} << {Gamma2} means that the union of all elements of {Gamma1} is a subset of the union of all elements of {Gamma2}, while the intersection of all elements of {Gamma2} is a subset of the intersection of all elements of {Gamma1}. Let us say that a family {Gamma} is a Konig-Egervary collection if f(Gamma) = 2*alpha(G). We conclude with the observation that for every graph G each subcollection of a Konig-Egervary collection is Konig-Egervary as well.

cs.DM↗
arXiv2015

Critical and Maximum Independent Sets of a Graph

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. By Ind(G) we mean the family of all independent sets of G while core(G) and corona(G) denote the intersection and the union of all maximum independent sets, respectively. The number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang, 1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of all critical independent sets of G (Levit and Mandrescu, 2012). In this paper, we present various connections between critical unions and intersections of maximum independent sets of a graph. These relations give birth to new characterizations of Koenig-Egervary graphs, some of them involving ker(G), core(G), corona(G), and diadem(G).

cs.DM↗
arXiv2014

An AEC satisfying the disjoint amalgamation property, has arbitrarily large models

We study AECs without assuming the amalgamation property in general. We do assume the disjoint amalgamation property in a specific cardinality lambda and assume that there is no maximal model in λ. Under these hypotheses, we prove the following: 1. for every model, M, of cardinality λ, and every μ>λ, we can find a model M^* of cardinality μ, extending M. 2.(λ,λ,μ)-amalgalmation property: for every three models M,N,M^* of cardinalities λ,λ,μ, respectively, if M<M^* and M<N then we can amalgamate M^* and N over M.

math.LO↗
arXiv2011

Independence of Sets Without Stability

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.

math.LO↗
arXiv2011

Weakening the local character

In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K,\preceq), with LST-number at most λ, which is categorical in λand λ^+ and has less than 2^{λ^+} models of cardinality λ^{++}, but at least one. This non-forking relation satisfies the main properties of the non-forking relation on stable first order theories, but only a weak version of the local character. Here, we improve this non-forking relation such that it satisfies the local character, too. Therefore it satisfies the main properties of the non-forking relation on superstable first order theories. We conclude that the function λ\to I(λ,K), which assigns to each cardinal λ, the number of models in K of cardinality λ, is not arbitrary.

math.LO↗
arXiv2010

Good Frames With A Weak Stability

Let K be an abstract elementary class of models. Assume that there are less than the maximal number of models in K_{λ^{+n}} (namely models in K of power λ^{+n}) for all n. We provide conditions on K_λ, that imply the existence of a model in K_{λ^{+n}} for all n. We do this by providing sufficiently strong conditions on K_λ, that they are inherited by a properly chosen subclass of K_{λ^+}.

math.LO↗