arXiv subjects

Menachem Kojman

Publications and source records attributed to Menachem Kojman.

29 records · Page 2Linked to original sources

arXiv1995

Representing embeddability as set inclusion

A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graph's end-structure. Using a combinatorial theorem of Shelah it is proved: - The complexity of the class in every regular uncountable $ł>\aleph_1$ is at least $λ^+ + \sup\{μ^{\aleph_0}:μ^+<λ\}$ - For all regular uncountable $λ>\aleph_1$ there are $2^λ$ pairwise non embeddable graphs in the class having strong homogeneity properties. - It is characterized when some invariants of a graph $G\in \Cal G_λ$ have to be inherited by one of fewer than $λ$ subgraphs whose union covers $G$. All three results are obtained as corollaries of a representation theorem that asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality $λ>\aleph_1$ onto set inclusion over all subsets of reals or cardinality $λ$ or less. Continuity properties of the homomorphism are used to extend the first result to all singular cardinals below the first cardinal fixed point of second order. The first result shows that, unlike what Shelah showed in the class of all graphs, the relations of embeddability in this class is not independent of negations of the GCH.

math.LO↗

arXiv1995

On universal graphs without cliques or withour large bipartite graphs

For every uncountable cardinal $λ$, suitable negations of the Generalized Continuum Hypothesis imply: - For all infinite $α$ and $β$, there is no universal $K_{α,β}$-free graphs in $λ$ - For all $α\ge 3$, there is no universal $K_α$-free graph in $λ$ The instance $K_{ω,ω_1}$ for $λ=\aleph_1$ was settled by Komjath and Pach from the principle $\diamondsuit(ω_1)$.

math.LO↗

arXiv1995

A proof of Shelah's recent partition theorem

Shelah's own proof to his recent polarized partition theorem involving a singular strong limit that violates the GCH is presented. The proof is slightly re-arranges so that no use of the ideal I[λ] is made. The proof should be readabel to any student with knowledge of set theory.

math.LO↗

arXiv1994

Universal abelian groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a purely universal separable $p$-group in $\aleph_n$ if, and only if, $\cont\le \aleph_n$.

math.LO↗

arXiv1994

Infinite homogeneous bipartite graphs with unequal sides

We call a bipartite graph {\it homogeneous} if every finite partial automorphism which respects left and right can be extended to a total automorphism. A $(κ,λ )$ bipartite graph is a bipartite graph with left side of size $κ$ and right side of size $λ$. We show, using a theorem of Hrushovski on finite graphs, that there is a homogeneous $({\aleph_0},2^{\aleph_0} )$ bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending on the underlying set theory all homogeneous $({\aleph_0},\aleph_1)$ bipartite graphs may be isomorphic, or there may be $2^{\aleph_1}$ many isomorphism types of $(\aleph_0,\aleph_1)$ homogeneous graphs.

math.LO↗

arXiv1994

Homogeneous families and their automorphism groups

A homogeneous family of subsets over a given set is one with a very ``rich'' automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of $2^\cont$ isomorphism types of homogeneous families over a countable set.

math.LO↗

arXiv1994

Universal bridge-free graphs

We prove that there is no countable universal $B_n$-free graph for all $n$ and that there is no countable universal graph in the class of graphs omitting all cycles of length at most $2k$ for $k\ge 2$.

math.LO↗

arXiv1993

mu-complete Suslin trees on mu^+

We prove that mu = mu^{< mu}, 2^mu = mu^+ and ``there is a non reflecting stationary subset of mu^+ composed of ordinals of cofinality < mu'' imply that there is a mu-complete Souslin tree on mu^+ .

math.LO↗

arXiv1992

$μ$-complete Souslin trees on $μ^+$

We prove that $μ=μ^{<μ}$, $2^μ=μ^+$ and ``there is a non reflecting stationary subset of $μ^+$ composed of ordinals of cofinality $<μ$'' imply that there is a $μ$-complete Souslin tree on $μ^+$.

math.LO↗

arXiv1992

Non-existence of Universal Orders in Many Cardinals

Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<ł<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $ł$ and its existence is not a result of a trivial cardinal arithmetical reason, then $ł$ ``resembles'' $\aleph_1$ --- a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

math.LO↗

arXiv1992

The universality spectrum of stable unsuperstable theories

It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .

math.LO↗