Asymptotics for the logarithm of the number of $(k,l)$-solution-free collections in an interval of naturals

A collection $(A_1,\dots,A_{k+l})$ of subsets of an interval $[1,n]$ of naturals is called $(k,l)$-solution-free if there is no set $(a_1,\dots,$ $a_{k+l})\in A_1\times\dots\times A_{k+l}$ that is a solution to the equation $x_1+\dots+x_k=x_{k+1}+\dots+x_{k+l}$. We obtain the asymptotics for the logarithm of the number of sets $(k,l)$-free of solutions in an interval $[1,n]$ of naturals. Bibliogr. 17.