Combinatorics related to the continuum

我们研究了REAL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P(ω)/FIN这样的结构。(1)分配数。利用关于滤子F强迫L_F的有限支撑迭代，证明了η(P(ω)/FIN×P(ω)/FIN)&lt；η(C^ω/FIN)的相合性，其中η(A)是布尔代数A的分配数，C是科恩代数.这回答了道琼斯的一个问题。(2)GroupWise密度数字。设g是GroupWise密度数，g_f是理想的GroupWise密度数。证明了g&lt；g_f的相合性，从而回答了Mildenberger的一个问题。(3)拓扑群。设G=([ω]^&lt；&lt；ω&gt；，Δ)是自然数ω的有限子集的群，以对称差Δ为群乘。在与Michael Hrusak的共同工作中，我们得到了所有ω_1生成的滤子F在ω上的陈述的一致性，G上对应于F的群拓扑不是Frechet的。(4)强迫理论。利用CCC强迫的一种新的迭代技巧，我们得到了Shelah结果的一个新的证明，即u&lt；a是一致的，其中u是超滤子数，a是几乎不交数。(5)具有强组合性质的Mad族。在连续统假设CH下，我们与Greg Piper共同构造了同时是ο集的极大几乎不交(MAD)族，以及集中在可数子集上的MAD族。这证实了A.Miller的两个猜想。(6)类积模型的齐性性质。在与Sakae Fuchino的合作中，我们研究了几个在允许许多自同构的偏序的一般扩张中成立的组合原理。特别地，我们证明了齐性原理HP(κ)蕴含了Juhasz，Soukup和Szentmiklossy的组合原理C^S(κ)，并证明了HP(N_2)在类似科恩模型的产品模型中成立。

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.