Behavior for interval lengths c ≥ 3

Determine, for each real c ≥ 3 and each positive integer m, the largest integer f_c(m) defined as follows: f_c(m) is the maximum r such that for every set A = {a_1 < ··· < a_m} of m positive integers with maximum a_m and every real x, there exist r disjoint pairs (a, b) with a ∈ A, b an integer satisfying x < b < x + c a_m, and a dividing b. In other words, determine the exact value of f_c(m) (or tight bounds characterizing its growth in m) for fixed c ≥ 3.

Background

The paper solves Erdős Problem #650 by determining f(m) exactly when the interval length is 2a_m (and, more generally, for any c with 2 ≤ c < 3). It also explains that when 1 < c < 2 the analogous quantity collapses to 1 for all m.

However, for c ≥ 3 the authors note that the situation becomes much more complicated and they do not know what the correct behavior should be. They mention a known lower bound for c = 3 (that any interval of length 3a_m contains more than √(6m) distinct multiples of the a_i), but no general determination or tight characterization is known in this regime.