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Lecture Series ``Collegium Logicum''
Kurt Goedel Society
University of Technology, Vienna, Austria

Oesterreichische Mathematische Gesellschaft


Lecture Announcement
--------------------


MENACHEM KOJMAN
===============
Carnegie Mellon University, Pittsburgh
[kojman@hillgrove.math.cmu.edu]



Convexity, Chromatic Numbers and Set Theory
-------------------------------------------
Wednesday, May 4, 1994, 17:15

To every set  S  in a Euclidean space one can naturally associate
several hypergraphs which code the convexity properties of  S.  The
chromatic numbers of those hypergraphs convey certain decomposition
properties of  S.  For example, the chromatic number of the
d+1-dimensional hypergraph associated to S < R^d is the least
number of convex subsets needed to cover  S.

We shall discuss the interrelations between convex geometry of
sets in  R^d  and those chromatic numbers, for general sets and
for closed sets.  Results by M. Perles, S. Shelah and the speaker
that show that for dimension 3 or higher the connections are quite
loose. 


PLACE:
Freihaus HS 4, Wiedner Hauptstrasse 8-10, Turm B (gelber Bereich),
2. OG, 1040 Wien.



Homogeneous Bipartite Graphs and their Automorphism Groups
----------------------------------------------------------
Thursday, May 5, 1994, 15:00

The homogeneity of the countable random graph \Gamma, or  
Rado's graph, as it is sometimes called, is the following fact: any  
isomorphism between two finite induced subgraph of \Gamma extends  
to a total automorphism. Any graph satisfying this property is  
homogeneous, and all countable homogeneous graphs were classified by  
Lachlan and  Woodrow in the seventies. We look at homogeneous   
*bipartite graphs*. More precisely, suppose we restrict ourselves to  
automorphism of bipartite that preserve sides. The following  
local/global type question arises: Are the two sides of a homogeneous  
bipartite graphs necessarily of equal cardinalities?  A negative  
answer to this question was first given by M. Goldstern and the  
speaker, who constructed a counter example. We shall present this  
result and further results concerning the number of isomorphism types  
of homogeneous graphs with unequal sides. 


PLACE:
Seminarraum SEM 118, Institut fuer Algebra,
Wiedner Hauptstrasse 8-10, Turm A (gruener Bereich), 5. OG, 1040 Wien.

 

Universal Models in Uncountable Powers
--------------------------------------
Thursday, May 5, 1994, 16:45

We say that a class of models  K  (e.g. all models of a complete first
order theory T) has a universal model in cardinality  \lambda  if there
is a model  M \in K of cardinality  \lambda  with the property that
every model  N\in K  of cardinality  \lambda  is embedded in  M. Thus
the ordering of the rational numbers is universal in cardinality
\aleph_0 for the class of linear orderings.  The existence of
universal models in uncountable powers is often independent of
set theory and even of set theory with additional assumptions 
on cardinal arithmetics. Combinatorial tools developed by S. Shelah
enables a representation theorem for the embeddability relation
in certain classes in uncountable cardinalities.  We shall present
the following result: (joint work with S. Shelah):

There is a homomorphism from the relation of embeddabitity on linear
orderings of size  \aleph_2  to a ``simple'' partial order P. As a
corollary, if  \aleph_2 < 2^\aleph_0 then there is no universal
linear ordering of size  \aleph_2.


PLACE:
Seminarraum SEM 181A, Institut fuer Informationssysteme,
Paniglgasse 16, 1. Stock, 1040 Wien.


MENACHEM KOJMAN is currently visiting
Assistant Professor at the Department of Mathematics
at Carnegie Mellon University, Pittsburgh.
He can be reached by email at: kojman@hillgrove.math.cmu.edu.

------------------------
Kurt-Goedel-Gesellschaft
Technische Universitaet Wien
Institut fuer Computersprachen E185.2
Resselgasse 3/1, A-1040 Wien

Tel: (+43 1) 588 01/4088
Fax: (+43 1) 504 15 89
email: kgs@logic.tuwien.ac.at

Mit Unterstuetzung des
Bundesministeriums fuer Wissenschaft und Forschung