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                  ERDOS TALKS  2005
 
                   Endre Szemeredi
 
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Shalom all,
 
This year's Erdos talks will be delivered by Endre
Szemeredi, one of the most prominent combinatorists ever,
whose work is greatly influenced by Erdos. The topics he
has chosen for his talks are intended to reflect this to
some extent.
 
The first two talks will be delivered in the HUJI
combinatorics and CS theory seminars in Jerusalem, and the
third talk will be a plenary talk at the annual IMU meeting
in Neve Ilan.  Please find below the details:
 
SECOND LECTURE

Wednesday, May 18, 10:30-12:00, Math 110
            "A hypergraph analogue of Dirac's theorem on
             Hamiltonian cycles."
 
Talk 3:  Thursday, May 19, 16:25-17:10, Neve Ilan
             "Some Erdös-type problems in combinatorial number
              theory."
 
Abstracts:
 
Talk 2:
 
Dirac's theorem states that if in a graph G on n vertices
the degree of every vertex is at least n/2 then the graph
contains a Hamiltonian cycle.
 
In a 3-uniform hypergraph H, the co-degree of two vertices
x, y, denoted by d(x,y)  is the number of vertices z such
that {x,y,z} is an edge in H.
 
We are going to prove that if d(x, y) is at least n/2 for
every x and y, then H contains a Hamiltonian cycle, meaning
a cyclical ordering of the vertices such that every three
consecutive vertices form an edge. We will also prove the
analogous statement for k-uniform hypergraphs with k>3.
 
This is joint work with R. Rucinski and V. Rodl.
 
Talk 3:
 
In this talk we are going to discuss several problems
concerning the size, density, or structure of sets of
integers with certain properties. The approaches we use
will combine ideas and tools from combinatorics, number
theory, geometry, and harmonic analysis.
 
Below is a list of some of the problems that we wish to address.
 
a) Determine the minimum density of an infinite sequence of
integers A = a1, a2, a3 ... for which the collection of
finite subset sums of A contains an infinite arithmetic
progression.
 
b) For a finite set A and integer k, describe the structure
of kA, the set of all sums of k distinct elements in A.
 
c) For a finite set A, give a "good" bound on max {|A+A|,
|AA|}, where A+A is the set of all sums of two elements in
A and AA the set of all such products.
 
d) For a finite set A, give a "good" bound on max |B| where
B is a subset of A for which B+B is disjoint from A.
 
This is joint work with B. Sudakov and V. H. Vu.
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel <techm@math.technion.ac.il>
Announcement from: Ron Adin <radin@macs.biu.ac.il>