Induced lines in Hales-Jewett cubes

A line in [n]d is a set {x(1), . . . , x(n)} of n elements of [n]d such that for each 1 6 i 6 d, the sequence x i , . . . , x (n) i is either strictly increasing from 1 to n, or strictly decreasing from n to 1, or constant. How many lines can a set S ⊆ [n]d of a given size contain? One of our aims in this paper is to give a counterexample to the Ratio Conjecture of Patashnik, which states that the greatest average degree is attained when S = [n]d. Our other main aim is to prove the result (which would have been strongly suggested by the Ratio Conjecture) that the number of lines contained in S is at most |S|2−ε for some ε > 0. We also prove similar results for combinatorial, or Hales-Jewett, lines, i.e. lines such that only strictly increasing or constant sequences are allowed.