A Mad Q-set

可计算函数的最大最终不同族

Maximal Eventually Different Families of Computable Functions

研究可计算函数的最大最终不同（MED）族的质量问题，通过定义MED族质量问题并与Lempp等人定义的比较，借助Greenberg等人的框架，展示Schrittesser构建的有效封闭MED族，得出相关结果。

Definability and almost disjoint families

We show that there are no infinite maximal almost disjoint ("mad") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at $κ<2^{\aleph_0}$, then no $κ$-Souslin infinite almost disjoint family can be maximal. Finally we show that if $\aleph_1^{L[a]}<\aleph_1$, then there are no $Σ^1_2[a]$ infinite mad families.

Higher random indestructibility of MAD families

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $κ$ is indestructible by the higher random forcing $\mathbb Q_κ$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_κ) = \mathfrak b_κ= \mathfrak c_κ$ implies the existence $\mathbb Q_κ$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.

On two problems of Erdos and Hechler: New methods in singular Madness

For an infinite cardinal mu, MAD(mu) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over mu. Erdos and Hechler proved the consistency of [mu in MAD(mu)] for a singular cardinal mu and asked if it was ever possible for a singular mu that [mu notin MAD(mu)], and also whether 2^{cf(mu)}<mu ===> [mu in MAD(mu)] for every singular cardinal mu. We introduce a new method for controlling MAD(mu) for a singular mu and, among other new results about the structure of MAD(mu) for singular mu, settle both problems affirmatively.

向量子空间的Mad族与实数的最小非贫集

Mad families of vector subspaces and the smallest nonmeager set of reals

本文证明参数化菱形原理（对应贫理想一致性）蕴含可数向量空间中无限极大几乎不相交块子空间族的最小基数为阿列夫一，进而在Miller模型中该不变量为阿列夫一。

Madness in vector spaces

We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the "spectrum" of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on $ω$ . We apply the author's local Ramsey theory for vector spaces to give partial results concerning their definability.

The Ramsey property implies no mad families

We show that if all collections of infinite subsets of $\N$ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation $E_0$, and thus is constant on a "large" set. Furthermore we announce a number of additional results about mad families relativ…

A co-analytic maximal set of orthogonal measures

We prove that if $V=L$ then there is a $Π^1_1$ maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj that no analytic set of measures can be maximal orthogonal.

A note on Maximality of ideal-independent sets

In this note we derive a property of maximal ideal-independent subsets of boolean algebras which has corollaries regarding the continuum cardinals p and s_mm(P(omega)/fin).

Ramsey性质与高维mad族

The Ramsey property and higher dimensional mad families

本文在Ramsey正则性原则下证明，关于超限迭代Fréchet理想不存在无限极大几乎不交族，相关结果此前已在PNAS公布。

A module structure over maximal consistent sets

It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this provides the possibility to do some linear algebra on proofs. It will also be presented an algorithmic proceeding in order to deal with these particular type of deductions.

好的投射见证

Good projective witnesses

本文提出一种新的力迫概念，证明极大余有限群的最小基数可严格介于ℵ₁与𝔠之间，且存在最优的Π¹₂-可定义MCG，并在MAD族中建立了类似结果。

The Approachability Ideal Without a Maximal Set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $ω_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[ω_2]$ does not have a maximal set modulo clubs.

Computing Maximal Chains

In 1967 Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk's original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal …

Maximal clones on uncountable sets that include all permutations

We first determine the maximal clones on a set X of infinite regular cardinality which contain all permutations but not all unary functions, extending a result of Heindorf's for countably infinite X. If |X| is countably infinite or weakly compact, this yields a list of all maximal clones containing the permutations since in that case the maximal clones above the unary functions are known. We then generalize a result of Gavrilov's to obtain on all infinite X a list of all maximal submonoids of the monoid of unary functions which contain the permutations.