Abstract

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every, if P is a poset and the dimension of a subposet B of P is at most d whenever the cover graph of B is a block of the cover graph of P, then the dimension of P is at most d + 2.We also construct examples that show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.

Introduction

We assume that the reader is familiar with basic notation and terminology for partially ordered sets (here we use the short term posets), including chains and antichains, minimal and maximal elements, linear extensions, order diagrams, and cover graphs. Extensive background information on the combinatorics of posets can be found in [17, 18].

We will also assume that the reader is familiar with basic concepts of graph theory, including the following terms: connected and disconnected graphs, components, cut vertices, and k-connected graphs for an integer. Recall that when G is a connected graph, a connected induced subgraph H of G is called a block of G when H is 2-connected and there is no subgraph of G which contains H as a proper subgraph and is also 2-connected.

Here are the analogous concepts for posets. A poset P is said to be connected if its cover graph is connected. A subposet B of P is said to be convex if y ∈ B whenever x, z ∈ B and x < y < z in P. Note that when B is a convex subposet of P, the cover graph of B is an induced subgraph of the cover graph of P. A convex subposet B of P is called a component of P when the cover graph of B is a component of the cover graph of P. A convex subposet B of P is called a block of P when the cover graph of B is a block in the cover graph of P.

Abstract

This chapter describes work of Erdőss, Klarner, and Rado on semigroups of integer affine maps and on sets of integers they generate. It gives the history of problems they studied, some solutions, and new unsolved problems that arose from them.

Introduction

This chapter describes the history of an Erdőss problem on iteration of integer affine functions, gives its solution, and tours some related work. An integer affine function of one variable is a function of the form f (x) = mx + b for integers m, b. The Erdőss problem concerns the structure of integer orbits of a particular finitely generated semigroup of integer affine functions, with the semigroup operation being composition of maps.

In the early 1970s David Klarner and Richard Rado studied integer orbits of semigroups of such affine functions in an arbitrary number of variables, motivated by the work of Crampin and Hilton on self-orthogonal Latin squares described in the text that follows. In response to a question they posed about a particular example, Paul Erdőss proved a theorem on the size of an orbit for certain semigroups of univariate functions, upper bounding the number of integers below a given cutoff T occurring in such orbits, cf. [37, Theorem 8]. Erdős's interest in this orbit problem led him to offer a reward for a particular semigroup iteration problem. This problem was solved by Crampin and Hilton in 1972, but their solution was never published. We supply a reconstructed solution here.

We also present a history of selected later developments, including work of Mike Fredman, Don Knuth, David Klarner, and Don Coppersmith. Their work addresses the structure of particular affine integer semigroups, and sufficient conditions for an integer affine semigroup to be freely generated. The latter topic led Klarner to pose in 1982 several easy-to-state problems in the spirit of Erdős's prize problem, given at the end of the chapter, which remain unsolved.