Two New Extensions of the Hales-Jewett Theorem

We prove two extensions of the Hales-Jewett coloring theorem. The first is a polynomial version of a finitary case of Furstenberg and Katznelson’s multiparameter elaboration of a theorem, due to Carlson, about variable words. The second is an “idempotent” version of a result of Carlson and Simpson. MSC2000: Primary 05D10; Secondary 22A15. For k,N ∈ N, let W k denote the set of length-N words on the alphabet {0, 1, · · · , k− 1}. A variable word overWN k is a word w(x) of length N on the alphabet {0, 1, · · · , k− 1, x} in which the letter x appears at least once. If w(x) is a variable word and i ∈ {0, 1, . . . , k − 1}, we denote by w(i) the word that is obtained by replacing each occurrence of x in w(x) by an i. The Hales-Jewett theorem states that for every k, r ∈ N, there exists N = N(k, r) ∈ N such that for any partition WN k = ⋃r i=1 Ci, there exist j, 1 ≤ j ≤ r, and a variable word w(x) over WN k such that { w(i) : i ∈ {0, 1, . . . , k − 1} } ⊂ Cj . 1. Finitary extensions. In [BL], V. Bergelson and A. Leibman provided a “polynomial” version of the HalesJewett theorem. In order to formulate their result, we must develop some terminology. Let l ∈ N. A set-monomial (over N) in the variable X is an expression m(X) = S1 × S2 × · · · × Sl, where for each i, 1 ≤ i ≤ l, Si is either the symbol X or a nonempty singleton subset of N (these are called coordinate coefficients). The degree of the monomial is the number of times the symbol X appears in the list S1, · · · , Sl. For example, taking l = 3, m(X) = {5} × X × X is a set-monomial of degree 2, while m(X) = X × {17} × {2} is a set-monomial of degree 1. A set-polynomial is an expression of the form P (X) = m1(X) ∪ m2(X) ∪ · · · ∪ mk(X), where k ∈ N and m1(X), · · · ,mk(X) are set-monomials. The degree of a set-polynomial is the largest degree of its set-monomial “summands”, and its constant term consists of the “sum” of ∗The author acknowledges support from the National Science Foundation via a post doctoral fellowship administered by the University of Maryland. the electronic journal of combinatorics 7 (2000),#R49 2 those mi that are constant, i.e. of degree zero. Finally, we say that two set polynomials are disjoint if they share no set-monomial summands in common. Let F(S) denote the family of non-empty finite subsets of a set S. Any nonempty set polynomial p(A) determines a function from F(N) to F(Nl) in the obvious way (interpreting the symbol × as Cartesian product and the symbol ∪ as union). Notice that if P (X) and Q(X) are disjoint set-polynomials and B ∈ F(N) contains no coordinate coefficients of either P or Q then P (B) ∩Q(B) = ∅. Here now is the Bergelson-Leibman coloring theorem. Theorem 1.1. Let l ∈ N and let P be a finite family of set-polynomials over N whose constant terms are empty. Let I ⊂ N be any finite set and let r ∈ N. There exists a finite set S ⊂ N, with S ∩ I = ∅, such that if F (⋃ P∈P P (S) ) = ⋃r i=1Ci then there exists i, 1 ≤ i ≤ r, some non-empty B ⊂ S, and some A ⊂ ⋃ P∈P P (S) with A ∩ P (B) = ∅ for all P ∈ P and { A ∪ P (B) : P ∈ P } ⊂ Ci. Although the “polynomial” nature of Theorem 1.1 is at once clear, it is not immediately obvious that it includes the Hales-Jewett theorem as a special case, so we shall give a different formulation, and derive it from Theorem 1.1. Let k,N, d ∈ N. We denote byMk (d) the set of all function φ : {1, 2, . . . , N}d → {0, 1, . . . , k − 1}. When d = 2, one may identify this with the set of N × N matrices with entries belonging to {0, 1, . . . , k− 1}, so in general we shall refer to the members of Mk (d) as matrices, even when d > 2. A variable matrix over Mk (d) is a function ψ : {1, 2, . . . , N}d → {0, 1, . . . , k− 1, x} for which x appears in the range. The support of ψ is the set ψ−1(x); that is, the set of locations in the matrix where the symbol x appears. If ψ is a variable matrix over Mk (d), ψ is said to be standard if its support has the form B for some B ⊂ {1, 2, . . . , N}. We shall also consider multi-variable matrices ψ : {1, 2, . . . , N}d → {0, 1, . . . , k − 1, x1, x2, . . . , xt}. In this case we require that all the xi appear in the range, and we call ψ(xi) the ith support of ψ. If ψ is a t-variable matrix then ψ gives rise, via substitution, to a function w(x1, . . . , xt) : {0, . . . , k − 1} →Mk (d), and we will often refer to this induced w instead of to ψ when dealing with variable matrices. We require the following nonconventional notion of addition of matrices. We will introduce this notion in the context of dimension 2, although the obvious analogs are valid in arbitrary dimension. Let w = (wij)i,j=1 and y = (yij) M i,j=1 be matrices (variable or otherwise). If there exist disjoint sets W and Y , whose union is {1, . . . ,M}2, such that wij = 0 for (i, j) ∈ W and yij = 0 for (i, j) ∈ Y , then we define w + y = (zij)i,j=1, where zij = wij if (i, j) ∈ Y and zij = yij if (i, j) ∈ W . If however there exists (i, j) ∈ {1, . . . ,M}2 such that wij 6= 0 6= yij then the sum w + y is undefined. Theorem 1.2 The following are equivalent: