Large-Scale Regularities of Lattice Embeddings of Posets
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- 作者：Jeffrey B. Remmel and S. Gill Williamson
- 关键词：lattice ; poset ; Ramsey theory
- 刊名：Order
- 出版年：1999
- 出版时间：1999
- 年：1999
- 卷：16
- 期：3
- 页码：245-260
- 全文大小：133 KB
文摘
Let N denote the set of natural numbers and let P =(Nk, [(P)\tilde]{\tilde P} be the directed-incomparability graph of P which is defined to be the graph with vertex set equal to Nk and edge set equal to the set of all (x, y) such that max(x) [(P)\tilde]{\tilde P}D denote the restrictions of P and [(P)\tilde]{\tilde P} to D. Points x [(P)\tilde]{\tilde P} relative to the lattice Nk, and an analogous notion of monotone interior with respect to [(P)\tilde]{\tilde P} or [(P)\tilde]{\tilde P}D. We wish to identify situations where most of these interior points are [(P)\tilde]{\tilde P}Ek is exposed and (2) there is a fixed set C [(P)\tilde]{\tilde P}D belonging to Fk has its monotone concealment in the set C. In addition, we show that if P1 =(Nk, 1),..., Pr =(Nk, 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 Pi. We note that the main point of (2) is that the bound kk 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.