Forcing with quotients

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.     

Forcing and antifoundation

It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as “corecursion”). As an application it is shown that the generic extensions of standard models of ZFCA are models of ZFCA again.     

Arithmetical Sacks Forcing

We answer a question of Jockusch by constructing a hyperimmune-free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented.The first author was supported in part by the Marsden Fund of New Zealand.The second author was supported by a postdoctoral fellowship from the New Zealand Institute for Mathematics and its Applications, NSF of China No.10471060 and No.10420130638.     

Forcing in admissible sets

Translated from Algebra i Logika, Vol. 29, No. 6, pp. 648–658, November–December, 1990.     

Forcing in Finite Structures

We present a simple and completely model-theoretical proof of a strengthening of a theorem of Ajtai: The independence of the pigeonhole principle from IΔ₀(R). With regard to strength, the theorem proved here corresponds to the complexity/proof-theoretical results of [10] and [14], but a different combinatorics is used. Techniques inspired by Razborov [11] replace those derived from Håstad [8]. This leads to a much shorter and very direct construction.     

Forcing highly connected subgraphs

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2^k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007     

Forcing and Differentiable Functions

We consider covering $\aleph_1 \times \aleph_1$ rectangles by countably many smooth curves, and differentiable isomorphisms between $\aleph_1$ -dense sets of reals.     

Forcing with Proper Classes

ＦｏｒｃｉｎｇｗｉｔｈＰｒｏｐｅｒＣｌａｓｓｅｓＬｉＮａ（ＤｅｐａｒｔｍｅｎｔｏｆＰｏｌｉｔｉｃｓ，ＨｅｎａｎＵｎｉｖｅｒｓｄｙ，Ｋａｉｆｅｎｇ，４７５００１）Ａｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒ，ｕｓｉｎｇｔｈｅｍｏｄｅｌＲΔ（Ｂ）－ａｇｅｎｅｒａ...     

Forcing notions in inner models

There is a partial order \({\mathbb{P}}\) preserving stationary subsets of ω ₁ and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω ₁ over V also collapses ω ₁ over \({V^{\mathbb{P}}}\) . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.     

Forcing matchings on square grids

Let G be a graph that admits a perfect matching. The forcing number of a perfect matching M of G is defined as the smallest number of edges in a subset S M, such that S is in no other perfect matching. We show that for the 2n × 2n square grid, the forcing number of any perfect matching is bounded below by n and above by n². Both bounds are sharp. We also establish a connection between the forcing problem and the minimum feedback set problem. Finally, we present some conjectures about forcing numbers in other graphs.     

Forcing absoluteness and regularity properties

For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over and the core model . We use our results to answer open questions from set theory of the reals.     

Forcing Linearity on Greedy Codes

Described in this paper are two different methods of forcing greedy codesto be linear over arbitrary finite fields. Both methods are generalizationsof the binary B-greedy codes as well as the triangular greedycodes over arbitrary fields. One method generalizes to arbitrary orderingsover arbitrary finite fields while the other method generalizes theB-greedy codes to linear codes over arbitrary finite fields.Examples of the first method are computed for triangular greedy codes. Theseexamples give codes similar to triangular codes from B-orderings.Both methods are shown to be substantially different.     

The Bounded Axiom A Forcing Axiom

We introduce the Bounded Axiom A Forcing Axiom (BAAFA). It turns out that it is equiconsistent with the existence of a regular ∑₂‐correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom (BPFA) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

FORCING BONDS OF A BENZENOID SYSTEM

ＦＯＲＣＩＮＧＢＯＮＤＳＯＦＡＢＥＮＺＥＮＯＩＤＳＹＳＴＥＭＺＨＡＮＧＦＵＪＩ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉａｍｅｎＵｎｉｖｅｒｓｉｔｙ，Ｘｉａｍｅｎ８５００４６，Ｃｈｉｎａ）ＬＩＸＵＥＬＩＡＮＧ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐ...     

Forcing axioms and the continuum hypothesis

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π₂-sentences over the structure (H(ω ₂), ∈, NS _ω1), in the sense that its (H(ω ₂), ∈, NS _ω1) satisfies every Π₂-sentence σ for which (H(ω ₂), ∈, NS _ω1) ? σ can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π₂-sentences over the structure (H(ω ₂), ∈, ω ₁) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $ . In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.     

Forcing matching numbers of fullerene graphs

The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.     

Forcing axioms and the Galvin number

Periodica Mathematica Hungarica - We study the Galvin property. We show that various square principles imply that the cofinality of the Galvin number is uncountable (or even greater than $$aleph...     

Forcing clique immersions through chromatic number

Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t+4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no independent set of size three contains $K_{2\lfloor n∕5\rfloor }$ as an immersion.