Mathematics Seminar at ICTP - this week

[Math]

M A T H E M A T I C S   S E M I N A R S   2009

Thursday,  26 February at 15:00 hrs

Dr. Gyan Prakash
(Universite' de Bordeaux 1, Talence, France)

"Sum-free sets, sets with small sumset and the clique number of random
Cayley graphs."

Abstract: A subset $A$ of an abelian group is said to be sum-free if  
there
is no solution of the equation $x+y =z$ with $x,y, z \in A.$

Given a subset $B$ of a finite abelian group $G$, the set $B+B$  
consists of
all those elements of $G$, which can be written as a sum of two  
elements of
$B$. We say that a set has small sumset if the cardinality of $B+B$  
is not
much larger in compare to the cardinality of $B.$

In a joint work with R. Balasubramanian and D.S. Ramana, we observe  
that in
certain groups, the question of counting the total number of sum-free  
sets
is related to the question of counting the number of sets with small  
sumset.

In another work, we obtain an upper bound for the number of sets with  
small
sumset. Using this one obtain an upper bound for the number of sum- 
free sets
in a finite abelian groups of "type III" which is substantially  
better than
the bound implied by the result of Ben Green and Imre Ruzsa.

Moreover using this, one also obtain an existence of Ramsey graphs among
Cayley graphs and thereby proving a weaker version of a conjecture of  
Noga
Alon.

Venue:  SEMINAR ROOM, (ICTP Leonardo Building, first floor)