Abstract There exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions of weakness. In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems.

Abstract A mass problem is a set of functions $\omega \to \omega $ . For mass problems ${\mathcal {C}}, {\mathcal {D}}$ , one says that ${\mathcal {C}}$ is Muchnik reducible to ${\mathcal {D}}$ if each function in ${\mathcal {C}}$ is computed by a function in ${\mathcal {D}}$ . In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility. For $p \in [0,1]$ let ${\mathcal {D}}(p)$ be the mass problem of infinite bit sequencesy(i.e., $\{0,1\}$ -valued functions) such that for each computable bit sequencex, the bit sequence $ x {\,\leftrightarrow\,} y$ has asymptotic lower density at mostp(where $x {\,\leftrightarrow\,} y$ has a $1$ in positionniff $x(n) = y(n)$ ). We show that all members of this family of mass problems parameterized by a realpwith $0 < p <1/2 $ have the same complexity in the sense of Muchnik reducibility. We prove this by showing Muchnik equivalence of the problems ${\mathcal {D}}(p)$ with the mass problem $\text {IOE}({2^{2}}^{n})$ ; here for an order functionh, the mass problem $\text {IOE}(h)$ consists of the functionsfthat agree infinitely often with each computable function bounded byh. This result also yields a new version of the proof to of the affirmative answer to the “Gamma question” due to the first author: $\Gamma (A)< 1/2$ implies $\Gamma (A)=0$ for each Turing oracleA. As a dual of the problem ${\mathcal {D}}(p)$ , define ${\mathcal {B}}(p)$ , for $0 \le p < 1/2$ , to be the set of bit sequencesysuch that $\underline \rho (x {\,\leftrightarrow\,} y)> p$ for each computable setx. We prove that the Medvedev (and hence Muchnik) complexity of the mass problems ${\mathcal {B}}(p)$ is the same for all $p \in (0, 1/2)$ , by showing that they are Medvedev equivalent to the mass problem of functions bounded by ${2^{2}}^{n}$ that are almost everywhere different from each computable function. Next, together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type $\text {IOE}$ : we show that for any order functiongthere exists a faster growing order function $h $ such that $\text {IOE}(h)$ is strictly above $\text {IOE}(g)$ in the sense of Muchnik reducibility. We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For ins

We answer in this paper an open question (known as the "Gamma question"), related to the recent notion of coarse computability, which stems from complexity theory. The question was formulated by Andrews, Cai, Diamondstone, Jockusch and Lempp in "Asymptotic density, computable traceability and 1-randomness" [1]. The Gamma value of an oracle set measures to what extent each set computable with the oracle is approximable in the sense of density by a computable set. The closer to 1 this value is, the closer the oracle is to being computable. The Gamma question asks whether this value can be strictly in between 0 and 1/2.

The Gamma question was formulated by Andrews et al. In "Asymptotic density, computable trace ability and 1-randomness" (2013, available at http://www.math.wisc.edu/~lempp/papers/traceable.pdf). It is related to the recent notion of coarse computability which stems from complexity theory. The Gamma value of an oracle set measures to what extent each set computable with the oracle is approximable, in the sense of density, by a computable set. The closer to 1 this value is, the closer the oracle is to being computable. The Gamma question asks whether this value can be strictly in between 0 and 1/2. We say that an oracle is weakly Schnorr engulfing if it computes a Schnorr test that succeeds on all computable reals. We show that each non weakly Schnorr engulfing oracle has a Gamma value of at least 1/2. Together with a recent result of Kjos-Hanssen, Stephan, and Terwijn, this establishes new examples of such oracles. We also give a unifying approach to oracles with Gamma value 0. We say that an oracle is infinitely often equal with bound h if it computes a function that agrees infinitely often with each computable function bounded by h. We show that every oracle which is infinitely equal with bound 2dn for d>1 has a Gamma value of 0. This provides new examples of such oracles as well. We present a combinatorial characterization of being weakly Schnorr engulfing via traces, which is inspired by the study of cardinal characteristics in set theory.

[Kechris, Trans. Amer. Math. Soc. 1975] showed that there exists a largest Pi_1^1 set of measure 0. An explicit construction of this largest Pi_1^1 nullset has later been given in [Hjorth and Nies, J. London Math. Soc. 2007]. Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned by Chong and Yu, and in [Yu, Fund. Math. 2011]). In this paper, we refute this conjecture and show that this nullset is merely Sigma_3^0. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Sigma_3^0. To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties.

Suppose you want to generate a random sequence of zeros and ones and all you have at your disposal is a coin which you suspect to be biased (but do not know the bias). Can "perfect" randomness be produced with this coin? The answer is positive, thanks to a little trick discovered by von Neumann. In this paper, we investigate a generalization of this question: if we have access to a source of bits produced according to some probability measure in some class of measures, and suppose we know the class but not the measure (in the above example, the class would be the class of all Bernoulli measures), can perfect randomness be produced? We will look at this question from the viewpoint of effective mathematics and in particular the theory of effective randomness.