James H. Schmerl

Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures

Abstract A finite, indecomposable partially ordered set is said to be critically indecomposable if, whenever an element is removed, the resulting induced partially ordered set is not indecomposable. The same terminology can be applied to graphs, tournaments, or any other relational structure whose relations are binary and irreflexive. It will be shown in this paper that critically indecomposable partially ordered sets are rather scarce; indeed, there are none of odd order, there is exactly one of order 4, and for each even k ⪖ 6 there are exactly two of order k . The same applies to graphs. For tournaments, there are none have even order, there is exactly one of order 3, and for each odd k ⪖ 5 there are precisely three of order k . In general, for arbitrary irreflexive binary relational structures, we will see that all critical indecomposables fall into one of nine infinite classes. Four of these classes are even—they contain no structures of odd order and for even k ⪖ 6 they each contain (up to a certain type of equivalence) exactly one structure of order k . The five other classes sre odd—they contain no structures of even order and for each odd k ⪖ 5 they each contain exactly one structure of order k . From this characterization of critically indecomposable structures, it will be evident that all indecomposable substructures of critically indecomposable structures are themselves critically indecomposable. Finally, it is proved that every indecomposable structure of order n + 2 ( n ⪖ 5) has an indecomposable substructure of order n .

Countable homogeneous partially ordered sets

It is shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2]. A classification of these partially ordered sets is given.

On the size of jump-critical ordered sets

The maximum size of a jump-critical ordered set with jump-number m is at most (m+1)!

Making the hyperreal line both saturated and complete

In a nonstandard universe, the K-saturation property states that any family of fewer than K internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the A-Bolzano-Weierstrass property iff F has cofinality A and every bounded A-sequence in F has a convergent A-subsequence. We show that if K < A are uncountable regular cardinals and fl, < A whenever a < K and /1 < A, then there is a K-saturated nonstandard universe in which the hyperreal numbers have the A-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second

A combinatorial result about points and balls in Euclidean space

For eachn≥1 there iscn>0 such that for any finite sexX ⊆ ℝ″ there isA ⊆X, |A|≤1/2(n+3), having the following property: ifB ⊇A is ann-ball, then |B ∩X|≥cn|X|. This generalizes a theorem of Neumann-Lara and Urrutia which states thatc2≥1/60.

On maximal subgroups of the automorphism group of a countable recursively saturated model of PA

Abstract We show that the stabilizer of an element a of a countable recursively saturated model of arithmetic M is a maximal subgroup of Aut(M) iff the type of a is selective. This is a point of departure for a more detailed study of the relationship between pointwise and setwise stabilizers of certain subsets of M and the types of elements in those subsets. We also show that a complete type of PA is 2-indiscernible iff it is minimal in the sense of Gaifman.

Decidability and 0 -Categoricity of Theories of Partially Ordered Sets.

This paper is primarily concerned with ℵ 0 -categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ 0 -categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ 0 - categorical there is a linear order of A such that ( , is ℵ 0 - categorical . Corollary 6.7. Every ℵ 0 - categorical theory of a partially ordered set of finite width has a decidable theory . Theorem 7.7. Every ℵ 0 - categorical theory of reticles has a decidable theory . There is a section dealing just with decidability of partially ordered sets, the main result of this section being Theorem 8.2. If (P, P is the class of partially ordered sets which do not embed (P, Th( K P ) is decidable iff K P contains only reticles .

The chromatic number of graphs which induce neither K 1,3 nor K 5 - e

Abstract If G is a graph which induces neither K 1,3 nor K 5 − e and if Δ ( G )⩽2 ω ( G )−5, then χ ( G ) = ω ( G ). Conversely, for each k ⩾ 4 there is a graph G which induces neither K 1,3 nor K 5 − e such that ω ( G ) = k , Δ ( G ) = 2 k − 3 and χ ( G ) = k + 1.

MODELS OF PEANO ARITHMETIC AND A QUESTION OF SIKORSKI ON ORDERED FIELDS

Using models of Peano Arithmetic, we solve a problem of Sikorski by showing that the existence of an ordered field of cardinalityλ with the Bolzano-Weierstrass property forκ-sequences is equivalent to the existence of aκ-tree with exactlyλ branches and with noκ-Aronszajn subtrees.

A partition property characterizing cardinals hyperinaccessible of finite type

Let P(n, (2) be the class of infinite cardinals which have the following property: Suppose for each v<KK that Cv is a partition of [KIn and card (C.)< K; then there is X C K of length a such that for each v < K, the set X (v + 1) is C,-homogeneous. In this paper the classes P (n, a) are studied and a nearly complete characterization of them is given. A principal result is that P (n + 2, n + 5) is the class of cardinals which are hyperinaccessible of type n. A partition property which differs from usual ones in that many partitions are considered simultaneously is defined and investigated in this paper. This property is interesting because it leads to an elementary characterization of the class of cardinals which are hyperinaccessible of a given finite type. The motivation for such a characterization comes from a problem in model theory. This problem is satisfactorily solved (as announced in [41 and [51) using some of the combinatorial results of this paper. These results lead naturally to a combinatorial problem which seems to be of sufficient independent interest so as to warrant further investigation. We give here an almost complete solution of this combinatorial problem; the final step for a complete solution seems elusive. This paper is a reworked version of Chapter 7 of my Ph. D. thesis [61 written under the supervision of Professor Robert L. Vaught. The other parts of my thesis, which consist of the model-theoretic applications of the results included here as well as generalizations of these results, will appear elsewhere. 1. The basic concepts. An ordinal number is the set of its predecessors. Ordinals are denoted by the Greek letters a, 3, y, v, e. Cardinal numbers are identified with initial ordinals and are denoted by K, A, li, where K is always an infinite cardinal. The symbols n, m, i always denote finite ordinals. A cardinal K is a strong limit cardinal iff 2\ < K whenever A < K. An inaccessible cardinal is a regular, strong limit cardinal. We need the concept of a Received by the editors June 17, 1971. AMS (MOS) subject classifications (1970). Primary 02K35, 04A10; Secondary 02H05, 02H13.