For a rational number r > 1, a set of positive integers is called an r-multiple-free set if A does not contain any solution of the equation rx = y. The extremal problem of estimating the maximum possible size of r-multiple-free sets contained in [n]:= {1, 2,...,n} has been studied in combinatorial number theory for theoretical interest and its application to coding theory. Let a and b be relatively prime positive integers such that a < b. Wakeham and Wood showed that the maximum size of (6/a)-multiple-free sets contained in [n] is b+1/b + O(logn). In this note we generalize this result as follows. For a real number p ∈ (0,1),let [n]p be a set of integers obtained by choosing each element i ∈ [n] randomly and independently with probability p. We show that the maximum possible size of (b/a)-multiple-free sets contained in [n]p is + log n log log n) with probability that goes to 1 as n →∞.
展开▼
