Induced lines in Hales–Jewett cubes

A line in d[n] is a set {x(1),…,x(n)} of n elements of d[n] such that for each 1⩽i⩽d, the sequence is either strictly increasing from 1 to n, or strictly decreasing from n to 1, or constant. How many lines can a set S⊆d[n] 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=d[n]. 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.