Inductive theories and their forcing companions

A decomposition theorem for ideals of a distributive lattice is related to a classification of the generic models of an arbitrary inductive theory, generalizing, for example, the classification of algebraically closed fields according to their characteristics. Research supported in part by the National Science Foundation Grant No. GP-29218.     

The upper connected geodetic number and forcing connected geodetic number of a graph

For a connected graph G of order p≥2, a set S⊆V(G) is a geodetic set of G if each vertex v∈V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g_c(G). A connected geodetic set of cardinality g_c(G) is called a g_c-set of G. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number is the maximum cardinality of a minimal connected geodetic set of G. We determine bounds for and determine the same for some special classes of graphs. For positive integers r,d and n≥d+1 with r≤d≤2r, there exists a connected graph G with , and . Also, for any positive integers 2≤a<b≤c, there exists a connected graph G such that g(G)=a, g_c(G)=b and . A subset T of a g_c-set S is called a forcing subset for S if S is the unique g_c-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected geodetic number of S, denoted by f_c(S), is the cardinality of a minimum forcing subset of S. The forcing connected geodetic number of G, denoted by f_c(G), is f_c(G)=min{f_c(S)}, where the minimum is taken over all g_c-sets S in G. It is shown that for every pair a,b of integers with 0≤a≤b−4, there exists a connected graph G such that f_c(G)=a and g_c(G)=b.     

The minimum forcing number of perfect matchings in the hypercube

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim (1998) conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is $2^{n?2}$, for all $n\ge 2$. This was revised by Riddle (2002), who conjectured that it is at least $2^{n?2}$, and proved it for all even $n$. We show that the revised conjecture holds for all $n\ge 2$. The proof is based on simple linear algebra.     

Algebraic properties of number theories

Among other things we prove the following. (A) A number theory is convex if and only if it is inductive. (B) No r.e. number theory has JEP. (C) No number theory has AP. We also give some information about the hard cores of number theories.     

The number of uncountable models of ω-stable theories

In this paper we give a complete answer to the question of determining all possible functions I(μ, T), where I(μ, T) denotes the number of non-isomorphic models of a fixed countable ω-stable theory T or cardinality μ (μ uncountable).     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.     

On the chromatic forcing number of a random graph

When we wish to compute lower bounds for the chromatic number χ(G) of a graph G, it is of interest to know something about the ‘chromatic forcing number’ f_χ(G), which is defined to be the least number of vertices in a subgraph H of G such that χ(H) = χ(G). We show here that for random graphs G_n,p with n vertices, f_χ(G_n,p) is almost surely at least ($\text{1}\text{2}$?ε)n, despite say the fact that the largest complete subgraph of G_n,p has only about log n vertices.     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Stable theories, pseudoplanes and the number of countable models

We prove that if T is a stable theory with only a finite number (>1) of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal (in the sense of [9]).     

The subcompleteness of Magidor forcing

It is shown that the Magidor forcing to collapse the cofinality of a measurable cardinal that carries a length \(\omega _1\) sequence of normal ultrafilters, increasing in the Mitchell order, to \(\omega _1\), is subcomplete.     

A technique for computing the zero forcing number of a graph with a cut-vertex

The zero forcing number of a graph is the minimum size of a zero forcing set. This parameter is useful in the minimum rank/maximum nullity problem, as it gives an upper bound to the maximum nullity. Results for determining graphs with extreme zero forcing numbers, for determining the zero forcing number of graphs with a cut-vertex, and for determining the zero forcing number of unicyclic graphs are presented.     

C-minimality and model companions

We study the question on existence of a model companion of the theory of a C-minimal structure with a distinguished automorphism of a special form.     

On the number of countable models of complete theories with finite Rudin-Keisler preorders

The aim of this article is to generalize the classification of complete theories with finitely many countable models with respect to two principal characteristics, Rudin-Keisler preorders and the distribution functions of the number of limit models, to an arbitrary case with a finite Rudin-Keisler preorder. We establish that the same characteristics play a crucial role in the case we consider. We prove the compatibility of arbitrary finite Rudin-Keisler preorders with arbitrary distribution functions f satisfying the condition rang f?ω∪{ω, 2^ω}.     

The gyroid is embedded and has constant mean curvature companions

The gyroid is a triply periodic minimal surface in the associated family of the SchwarzP- andD-surface. We prove it is embedded and find constant mean curvature companions of the gyroid with small constant mean curvature. We also discuss a surface similar to the gyroid in the associated family of the SchwarzH-surface.