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, $\alpha(F)$. A Konig Egervary (KE) collection is a relevant collection $F$, that satisfies $|\bigcup F|+|\bigcap F|=2\alpha(F)$. 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 $\Gamma$ be a relevant collection such that $\Gamma-\{S\}$ is an hke collection, for every $S \in \Gamma$. We study the difference between $|\bigcap \Gamma_1-\bigcup \Gamma_2|$ and $|\bigcap \Gamma_2-\bigcup \Gamma_1|$, where $\{\Gamma_1,\Gamma_2\}$ is a partition of $\Gamma$. We get new characterizations for an hke collection and for a KE graph.Comment: 6 Page

Duality and Hereditary K\"onig-Egerv\'ary Set-systems

A K\"onig-Egerv\'ary graph is a graph $G$ satisfying $\alpha(G)+\mu(G)=|V(G)|$, where $\alpha(G)$ is the cardinality of a maximum independent set and $\mu(G)$ is the matching number of $G$. Such graphs are those that admit a matching between $V(G)-\bigcup \Gamma$ and $\bigcap \Gamma$ where $\Gamma$ is a set-system comprised of maximum independent sets satisfying $|\bigcup \Gamma'|+|\bigcap \Gamma'|=2\alpha(G)$ for every set-system $\Gamma' \subseteq \Gamma$; in order to improve this characterization of a K\"onig-Egerv\'ary graph, we characterize \emph{hereditary K\"onig-Egerv\'ary set-systems} (HKE set-systems, here after). An \emph{HKE} set-system is a set-system, $F$, such that for some positive integer, $\alpha$, the equality $|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha$ holds for every non-empty subset, $\Gamma$, 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 $|\bigcap \Gamma_1-\bigcup \Gamma_2|=|\bigcap \Gamma_2-\bigcup \Gamma_1|$ holds for every two non-empty disjoint subsets, $\Gamma_1,\Gamma_2$ of $F$. This theorem is applied in \cite{hke},\cite{broken}.Comment: 6 page

Semi-Good Frames with Amalgamation and Tameness in lambda^+

We present a connection between tameness and non-forking frames. In addition we improve results about independence and dimension.Comment: arXiv admin note: substantial text overlap with arXiv:0901.085

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 $\alpha(G) + \mu(G)= |V(G)|$, 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. A collection $F$ of sets is an hke collection if $|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha$ holds for every subcollection $\Gamma$ 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 $\Omega(G)$ 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 $F$ is an hke collection if and only if it is a subset of $\Omega(G)$ for some graph $G$ and $|\bigcup F|+|\bigcap F|=2\alpha(F)$. Finally, we show that the maximal cardinality of an hke collection $F$ with $\alpha(F)=\alpha$ and $|\bigcup F|=n$ is $2^{n-\alpha}$.Comment: 20 Page

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 $\lambda^+$-frame from a semi-good non-forking $\lambda$-frame. But the classes $K_{\lambda^+}$ and $\preceq \restriction K_{\lambda^+}$ are replaced: $K_{\lambda^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{\lambda^+}$ is restricted to the partial order $\preceq^{NF}_{\lambda^+}$. Here, we avoid the restriction of the partial order $\preceq \restriction K_{\lambda^+}$, assuming that every saturated model (in $\lambda^+$ over $\lambda$) is an amalgamation base and $(\lambda,\lambda^+)$-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 $M \preceq^{NF}_{\lambda^+}M^+$, provided that $M$ and $M^+$ are saturated models. We present sufficient conditions for three good non-forking $\lambda^+$-frames: one relates to all the models of cardinality $\lambda^+$ and the two others relate to the saturated models only. By an `unproven claim' of Shelah, if we can repeat this procedure $\omega$ times, namely, `derive' good non-forking $\lambda^{+n}$ frame for each $n<\omega$ 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

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_{\lambda^{+n}} (namely models in K of power \lambda^{+n}) for all n. We provide conditions on K_\lambda, that imply the existence of a model in K_{\lambda^{+n}} for all n. We do this by providing sufficiently strong conditions on K_\lambda, that they are inherited by a properly chosen subclass of K_{\lambda^+}

Density of uniqueness triples from the diamond axiom

We work with a pre-$\lambda$-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 $\lambda$. We investigate the density of uniqueness triples in a given pre-$\lambda$-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 $\lambda^+$, and we do assume $\diamondsuit(\lambda^+)$. As a corollary, by applying our main result to the trivial $\lambda$-frame, it follows that in any AEC $\mathbf K$ satisfying modest hypotheses on $\mathbf K_\lambda$ and $\mathbf K_{\lambda^+}$, the set of $*$-domination triples in $\mathbf K_\lambda$ 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.Comment: Expanded with more corollaries since v2; now 37 page

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).Comment: 12 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1407.736

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.Comment: 15 pages, 7 figure

Weakening the local character

In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K,\preceq), with LST-number at most \lambda, which is categorical in \lambda and \lambda^+ and has less than 2^{\lambda^+} models of cardinality \lambda^{++}, 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 \lambda \to I(\lambda,K), which assigns to each cardinal \lambda, the number of models in K of cardinality \lambda, is not arbitrary.Comment: 11 page