Large-Scale Regularities of Lattice Embeddings of Posets

Abstract

Let N denote the set of natural numbers and let P =(N ^k, ≼) be a countably infinite poset on the k-dimensional lattice N ^k. Given x ∈ N ^k, we write max(x) (min(x)) for the maximum (minimum) coordinate of x. Let ${\tilde P}$ be the directed-incomparability graph of P which is defined to be the graph with vertex set equal to N ^k and edge set equal to the set of all (x, y) such that max(x) ≥ max(y) and x and y not comparable in P. For any subset D ⫅ N ^k, we let P _D and ${\tilde P}$ _D denote the restrictions of P and ${\tilde P}$ to D. Points x ∈ N ^k with min(x) = 0 will be called boundary points. We define a geometrically natural notion of when a point is interior to P or ${\tilde P}$ relative to the lattice N ^k, and an analogous notion of monotone interior with respect to ${\tilde P}$ or ${\tilde P}$ _D . We wish to identify situations where most of these interior points are “exposed” to the boundary of the lattice or, in the case of monotone interior points, not “concealed” very much from the boundary. All of these ideas restrict to finite sublattices F ^k and/or infinite sublattices E ^k of N ^k. Our main result shows that for any poset P and any arbitarily large integer M > 0, there is an F ⊂ E with ∣ F ∣ = M where, relative to the sublattices F ^k ⊂ E ^k, the ideal situation of total exposure of interior points and “very little” concealment of monotone interior points must occur. Precisely, we prove that for any P =(N ^k, ≼) and any integer M > 0, there is an infinite E ⫅ N and a finite D ⫆ F ^k with F ⊂ E and ∣ F ∣ = M such that (1) every interior vertex of P _{E k} or ${\tilde P}$ _{E k} is exposed and (2) there is a fixed set C ⊂ E, ∣ C ∣ ≤ k ^k, such that every monotone-interior point of ${\tilde P}$ _D belonging to F ^k has its monotone concealment in the set C. In addition, we show that if P ¹ =(N ^k, ≼₁),..., P ^r =(N ^k, ≼ _r ) is any sequence of posets, then we can find E,D, and F so that the properties (1) and (2) described above hold simultaneously for each P ⁱ. We note that the main point of (2) is that the bound k ^k depends only on the dimension of the lattice and not on the poset P. Statement (1) is derived from classical Ramsey theory while (2) is derived from a recent powerful extension of Ramsey theory due to H. Friedman and shown by Friedman to be independent of ZFC, the usual axioms of set theory. The fact that our result is proved as a corollary to a combinatorial theorem that is known to be independent of the usual axioms of mathematics does not, of course, mean that it cannot be proved using ZFC (we just couldn"t find such a proof). This puts our geometrically natural combinatorial result in a somewhat unusual position with regard to the axioms of mathematics.