Combinatorics related to the continuum

我们研究实数的组合结构及其与强迫理论的相互作用，以及集合论的其他领域。特别关注的是P(ω)/fin这样的结构。(1)分配性数。利用Laver强迫L_F对滤波器F的有限支持迭代，证明了η(P(ω)/fin x P(ω)/fin) < η(C^ω/fin)的一致性，其中η(a)是布尔代数a的分配数，C是科恩代数。这回答了陶氏的一个问题。(2)群密度数。设g为群密度数，设g_f为理想的群密度数。我们证明了g < g_f的一致性，从而回答了Mildenberger的一个问题。(3)拓扑群。设G = （[ω]^<<ω>， Δ）是具有对称差分Δ作为群乘法的自然数ω的有限子集群。在与Michael Hrusak的合作中，我们得到了所有ω_1生成的滤波器F在ω上的一致性，对应于F的G上的群拓扑不是Frechet。(4)强迫理论。利用一种新的ccc强迫迭代技术，我们得到了关于u < a是一致的Shelah结果的一个新的证明，其中u是超滤数，a是几乎不相交数。(5)组合性强的疯狂家族。在连续统假设CH下，我们构造了一个极大几乎不相交（mad）族，它既是一个ο集合，也是一个集中在可数子集上的疯狂族。这证实了米勒的两个猜想。(6)类产品模型的同质性。在与Sakae Fuchino的联合工作中，我们研究了几个组合原理，它们在允许许多自同构的偏序的一般扩展中成立。特别地，我们证明了同质性原理HP（κ）隐含了Juhasz、Soukup和Szentmiklossy的组合原理C^s(κ)，并证明了HP（N_2）在Cohen模型等产品模型中成立。

We investigate the combinatorial structure of the reals and its interplay with forcing theory, as well as other areas of set theory. Particular focus is put on structures like P(ω)/fin.(1) Distributivity numbers. Using finite support iteration of Laver forcing L_F with respect to a filter F, we prove the consistency of η(P(ω)/fin x P(ω)/fin) < η(C^ω/fin) where η(A) is the distributivity number of a Boolean algebra A, and C is the Cohen algebra. This answers a question of Dow.(2) Groupwise density numbers. Let g be the groupwise density number, and let g_f be the groupwise density number for ideals. We show the consistency of g < g_f, thus answering a question of Mildenberger.(3) Topological groups. Let G = ([ω]^<<ω>, Δ) be the group of finite subsets of the natural numbers ω equipped with symmetric difference Δ as group multiplication. In joint work with Michael Hrusak, we obtain the consistency of the statement for all ω_1-generated filters F on ω, the group topology on G corresponding to F is not Frechet.(4) Forcing theory. Using a novel iteration technique for ccc forcing, we obtain a new proof of Shelah's result saying that u < a is consistent where u is the ultrafilter number and a, the almost disjointness number.(5) Mad families with strong combinatorial properties. In joint work with Greg Piper, we construct, under the continuum hypothesis CH, a maximal almost disjoint (mad) family which is simultaneously a ο-set, as well as a mad family which is concentrated on a countable subset. This confirms two conjectures of A. Miller.(6) Homogeneity properties of product-like models. In joint work with Sakae Fuchino, we investigate several combinatorial principles which hold in generic extensions by partial-orders which admit many automorphisms. In particular, we show the homogeneity principle HP(κ) implies the combinatorial principle C^s(κ) of Juhasz, Soukup and Szentmiklossy, and we prove HP(N_2) holds in product models like the Cohen model.