On constant-multiple-free sets contained in random sets of integers

Abstract

For a rational number r > 1, a set A 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 (b/a)-multiple-free sets contained in [n] is b/b+1 n + O(log n).

In this note we generalize this result as follows. For a real number p is an element of (0, 1), let [n](p) be a set of integers obtained by choosing each element i is an element of [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 b/b+p pn + O(root pn log n log log n) with probability that goes to 1 as n -> infinity.