The Wildcard Set's Size in the Density Hales-Jewett Theorem

Let k,N ∈ N. We view members of {0, 1, . . . , k − 1} as words of length N on the alphabet {0, 1, . . . , k − 1}. A variable word is a word w1w2 · · ·wN on the alphabet {0, 1, . . . , k−1, x} in which the letter x (the variable) occurs. Indices i for which wi = x will be called wildcards, and {i : wi = x} will be called the wildcard set. We denote variable words by w(x), e.g. w(x) = 02x1x3210x is a variable word. If w(x) is a variable word and i ∈ {0, 1, . . . , k−1}, we denote by w(i) the word that results when all instances of “x” in w(x) are replaced by “i”. E.g. w(2) = 0221232102 for the variable word w(x) considered above. In [HJ], A. Hales and R. Jewett proved the following theorem.