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.
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