Question

What does it means "dissipative set" in the following context:

"If the set contains $3^n$ integer numbers, then it contains dissipative subset with $n$ elements"?

I have not found the definition related to the combinatoric...

Answer 1

Maybe, you mean "a dissociated set"? A finite subset $A$ of an abelian group is called dissociated if all of its $2^{|A|}$ subset sums are pairwise distinct. It is certainly true that if $A$ is set of $3^n$ elements of an (arbitrary) abelian group, then $A$ contains an $n$-element dissociated subset. Indeed, let $S$ be any maximal (by inclusion) dissociated subset of $A$. Then every element of $A$ can be written as a linear combination of the elements of $S$ with the coefficients in $\{-1,0,1\}$. Consequently, $3^{|S|}\ge|A|=3^n$, whence $|S|\ge n$.