Joseph G. Rosenstein

EFFECTIVE MATCHMAKING AND ¿-CHROMATIC GRAPHS

ALFRED B. MANASTER1 AND JOSEPH G. ROSENSTEIN2Abstract. In an earlier paper we showed that there is a recursivesociety, in which each person knows exactly two other people,whose marriage problem is solvable but not recursively solvable.We generalize this result, using a different construction, to the casewhere each person knows exactly k other people. From this wededuce that for each k^2 there is a recursive 2(Ar—l)-regular graph,whose chromatic number is k but which is not recursively k-chromatic.

Small recursive ordinals, many-one degrees, and the arithmetical difference hierarchy

Let W be the Markwald-Spector [ 10] set of GiSdel numbers for recur. sive well-ordering relations; for each recursive ordinal e, let W(a) denote the set of Gbdel numbers for recursive well-orderings of order type /1) and deg m W(a) the m-degree of W(t~). Then

Two-dimensional partial orderings: Recursive model theory

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings. The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈 A, R 〉 to be n -dimensional if there are n linear orderings of A , 〈 A, L 1 〉, 〈 A, L 2 〉 …, 〈 A, L n 〉 such that R = L 1 ∩ L 2 ∩ … ∩ L n . Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈 x 1 , y 1 〉 ≤ Q × Q 〈 x 2 , y 2 , if and only if x 1 ≤ x 2 and y 1 ≤ y 2 ) is 2-dimensional, since ≤ Q × Q is the intersection of the two lexicographic orderings of Q × Q . In fact, as shown by Dushnik and Miller, a countable partial ordering is n -dimensional if and only if it can be embedded as a subordering of Q n . Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

The Absence of Discrete Mathematics in Primary and Secondary Education in the United States… and Why that Is Counterproductive

This chapter describes the opportunity that discrete mathematics provides for supporting reasoning , problem solving , and systematic thinking in the school mathematics curriculum and illustrates this opportunity by providing a set of discrete mathematics problems that begin “Find all… .” It also provides a year-by-year model for how discrete mathematics can be included in the primary and secondary curriculum . Finally, the article describes some of the possible reasons why discrete mathematics was not included in the new national mathematics standards in the U.S., and why we consider these reasons misguided, in light of the opportunities provided when discrete mathematics is part of the curriculum.