Asymptotics for the Logarithm of the Number of ( k, l)-Solution-Free Collections in an Interval of Naturals

摘要

A collection ( A ~(1), … , 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), … , a ~( k + l )) ∈ A ~(1)× ? × A ~( k + l )that is a solution to the equation x ~(1)+ ? + x ~( k )= x ~( k +1)+ ? + 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.