[math-fun] Covering the first n naturals

Let 0 < d < 1 be a real number, and n be a positive integer. Let S be the set {1, 2, 3, ..., n}, and let T be a subset of S with cardinality CEIL[dn], where CEIL[] is the ceiling function. Does there exist D = f(d) such that S can be covered by D translated copies of T? If so, then we could D-colour S = {1, 2, 3, ..., n}, where each monochromatic subset is a translated copy of a subset of T. By van der Waerden's theorem, we have that (for sufficiently large n) there exists a monochromatic arithmetic progression of length k. This means that T (being a superset) must also contain such a monochromatic arithmetic progression, providing a proof of Szemeredi's theorem. Sincerely, Adam P. Goucher