arXiv subjects

Patrizio Cintioli

Publications and source records attributed to Patrizio Cintioli.

16 recordsLinked to original sources

arXiv2026

$\mathsf{RCA}_0$ Proves the Collapse of the Intersection Principles Hierarchy

This paper shows that the pairwise intersection principle ($2\mathrm{IP}$) implies, over $\mathsf{RCA}_0$, the existence of a Cohen $1$-generic real relative to every given set. Together with the known implications, this yields the equivalence over $\mathsf{RCA}_0$ of the finite intersection principle (FIP), $2\mathrm{IP}$, and $1$-GEN. In fact, FIP is equivalent to $n\mathrm{IP}$ for every fixed $n\geq 2$. These results give affirmative answers to Questions~1 and~2 of Cholak, Downey, and Igusa, and to Question~9.10.26 of Dzhafarov and Mummert. The proof uses cumulative, nested supports satisfying a finite Helly property and a decoding argument based on a least selected label. Neither $Σ^0_2$ induction nor $Σ^0_2$ bounding is required.

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arXiv2026

1-Genericity Is Not Weihrauch Reducible to the Finite Intersection Principle

This paper shows that the problem of producing a Cohen $1$-generic real relative to a given oracle is not Weihrauch reducible to the finite intersection principle (FIP). More strongly, no nontrivial computable FIP instance admits a single Turing functional that produces an absolute $1$-generic real from every enumeration of every maximal centered set of indices. The argument combines a $1$-generic real that computes no perfect tree all of whose paths are $1$-generic with local constancy and a parametrization of maximal centered sets. This answers Question~9.10.25 of Dzhafarov and Mummert.

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arXiv2026

Reconstruction of Finite Posets of Width Three

We prove that every finite ordered set of width at most three and cardinality at least four is reconstructible from its deck. This settles the width-three part of an open problem posed by Schröder in Ordered Sets; the separate question concerning ordered sets of height one is not addressed here. The new ingredient is a criterion for recovering the marked maximal deck of an ordered set with exactly three maximal elements from its unmarked maximal deck and the isomorphism type of the suborder obtained by deleting all maximal elements. We express the criterion as an orbital balance condition on the attachment ideals in this common base. Necessity follows by double counting; sufficiency uses the structure of automorphism actions in width at most three. The full-deck theorem then follows from reconstruction results of Schröder concerning maximal cards and marked maximal cards. A local reattachment argument correcting one intermediate step in the two-maximal-element case is included in an appendix.

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arXiv2026

Cone Avoidance and the No-Least-Join Theorem

We give a column-avoidance modification of the no-least-join construction of Downey, Greenberg, Lewis, and Montalbán. We show that if $C$ is computably enumerable, $A,B<_T C$, and $A\not\leq_T B$, then for every noncomputable set $E$ there is a set $X\leq_T C$ such that $B\leq_T A\oplus X$ and $E\not\leq_T X$. Consequently, for the corresponding Turing degrees $a,b,c$, the degrees $x\leq c$ satisfying $b\leq a\vee x$ have no nonzero common lower bound; equivalently, no nonzero Turing cone contains all of them. This gives a negative answer to their Question 1.19.

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arXiv2026

Pseudo-Hyperjump Inversion Fails for Turing Degrees

Jananthan and Simpson asked in Conjecture~4.5 whether every pseudo-hyperjump $\operatorname{HJ}_e(X)=X\oplus V_e^X$, with $V_e^X$ uniformly $Π^1_1(X)$, admits Turing-degree inversion above Kleene's $\mathcal{O}$. We settle their conjecture in the negative. We construct one index $e_*$ such that $X<_{\mathrm T}\operatorname{HJ}_{e_*}(X)$ for every real $X$, while the range of $\operatorname{HJ}_{e_*}$ omits every Turing degree $\mathbf d$ such that $°_{\mathrm T}(\mathcal{O})\leq\mathbf d<°_{\mathrm T}(\mathcal{O}^{\mathcal{O}})$. Thus Conjecture~4.5 fails even for an operator which strictly raises the Turing degree of every input. Each of Properties~4.6--4.8 implies Conjecture~4.5; consequently, the accompanying class-characterisation problems collapse: no class of reals, even without a definability assumption, satisfies any one of these properties.

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arXiv2026

A Special $Π^0_1$ Class with the Join Property but without Pseudojump Inversion

Jananthan and Simpson showed that special $Π^0_1$ classes exist in three of the four possible cases determined by whether the Join Property and the Pseudojump Inversion Property hold, leaving open the existence of a special $Π^0_1$ class with the Join Property but not the Pseudojump Inversion Property. We construct such a class, thereby answering their question affirmatively and completing the four-way existence pattern. Moreover, the class can be chosen to consist entirely of generalized low reals, with the Join Property holding locally on every nonempty relatively basic open subclass. More generally, every special $Π^0_1$ class contained in $\mathrm{GL}_1$ has a special $Π^0_1$ extension, still contained in $\mathrm{GL}_1$, on which the Join Property holds locally. We also show that no class contained in $\mathrm{GL}_1$ has the Pseudojump Inversion Property. An independent appendix proves that, for any class with the Join Property, every jump-inversion fibre above $0'$ is countably infinite.

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arXiv2026

A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees

In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles \textit{``Strong Reducibilities''} (1981) and \textit{``Reducibilities''} (1999). The problem asks whether every computably enumerable (c.e.) $tt$-degree contains a c.e.\ \textit{irreducible} $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\ $tt$-degree that does not contain any c.e.\ irreducible $m$-degree. Our proof relies on the structural properties of c.e.\ semirecursive sets with a rigid complement, originally constructed by A.~N.~Degtev. We show that the unique c.e.\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible $m$-degree within every c.e.\ $tt$-degree -- is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable.

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arXiv2026

Every introenumerable set contains a uniformly introreducible subset

We prove that every infinite introenumerable set contains an infinite uniformly introreducible subset, answering both parts of Question 1.7 of Greenberg, Harrison-Trainor, Patey, and Turetsky affirmatively. Moreover, a single procedure computes the original set from every infinite subset of the selected set. The proof combines the uniformization and refinement results of these authors with Ramsey-theoretic methods and a canonical reconstruction argument. We also obtain refinements controlling the ordinals and hyperjumps of the selected subsets.

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arXiv2026

A Bounded Finite-One Degree Whose One-One Degrees Form Exactly a Dense Linear Order

We construct a set $U\leq_T\emptyset''$ whose bounded finite-one degree, ordered internally by one-one reducibility, consists exactly of a countable dense linear order without endpoints. More precisely, \[ \left( \{[B]_1:B\equiv_{\mathrm{bfo}}U\},\leq_1 \right) \cong (\mathbb Q,\leq). \] This exact realization contrasts with two previous results. In earlier work, a bounded finite-one degree was constructed that contains a copy of $(\mathbb Q,\leq)$, but the same degree also contains an infinite antichain of one-one degrees and, more generally, embedded copies of all countable partial orders; thus the dense chain does not exhaust the degree. In a different direction, $m$-rigidity yields an almost-sure and comeager obstruction: for a measure-$1$ and comeager class of sets, the corresponding bounded finite-one degree contains an infinite antichain of one-one degrees and hence is not linearly ordered. The present construction shows that, despite this typical negative behaviour, exact dense linear order can occur. In particular, it gives an affirmative answer to Open Question~3 of Richter, Stephan, and Zhang. The proof has two main parts. A block homogenization construction produces a noncylindrical set $U$, a weak dyadic tower $(U_e)_{e\in\mathbb N}$, computable reservoirs of both colours, and a base absorption property. An abstract absorption-to-exhaustivity theorem then shows that every member of the bounded finite-one degree of $U$ is one-one equivalent to some finite autojoin $mU_e$, thereby yielding the exhaustive classification above.

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arXiv2026

A strengthened form of Dëgtev's argument and $D$-maximal many-one degrees

Dëgtev proved that if a noncomputable computably enumerable set $U$ satisfies his property $(R)$ and the many-one degree of $U$ contains no simple set, then the $1$-degree of $U$ is minimal. We show that his proof yields the stronger conclusion that, for every noncomputable $A\lem U$, \[ U\le_1 A. \] In particular, the injectivity of the reduction $A\le_1 U$ with which Dëgtev begins his proof is not needed. Using the Cholak--Gerdes--Lange classification, we show that for a noncomputable $D$-maximal set $D$, the hypothesis that its many-one degree contains no simple set is equivalent to $D$ being of one of Types~3--10. Hence, for every such $D$ and every noncomputable $A\lem D$, one has $D\le_1 A$. Together with Maslova's theorem for the simple-set cases, this recovers as a corollary the existence of a least finite-one degree in every noncomputable c.e.\ many-one degree containing a $D$-maximal set.

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arXiv2026

Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees

This paper proves that every nonrecursive many-one degree contains either exactly one or infinitely many finite-one degrees. This gives a negative answer to Open Question 2 of Richter, Stephan, and Zhang~\cite[p.~17]{RSZ}. Earlier work established that, for almost every $A\subseteq\N$ with respect to the standard product measure on $2^\N$, the degree $[A]_{\m}$ contains an infinite antichain of finite-one degrees~\cite{Cintioli}. Thus the question had already received an almost-sure negative answer; the present theorem settles it for every nonrecursive many-one degree.

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arXiv2026

A computably enumerable many-one degree with no least finite-one degree

Richter, Stephan, and Zhang asked whether every nonrecursive many-one degree contains a least finite-one degree. We solve this question in the negative, already within the class of computably enumerable many-one degrees. Positive answers are known in two disjoint natural settings: for a measure-one and comeager class of $m$-rigid sets, and, in a companion paper, for computably enumerable many-one degrees containing a $D$-maximal set. We construct a nonrecursive \ce\ set $A$ such that for every set $X \eqm A$ there exists a c.e.\ set $B \eqm A$ with $X \not\lfo B$. Hence the many-one degree of $A$ contains no least finite-one degree. The proof is a finite-injury priority construction based on virtual target sets and a dynamic trap mechanism forcing any putative finite-one reduction either to violate finite-oneness or to compute an incorrect reduction.

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arXiv2026

Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree

We construct a nonrecursive set $A\le_T\emptyset'$ and a uniformly computable family of sets $C_0,C_1,\dots$, all bounded finite-one equivalent to $A$, such that the corresponding $1$-degrees form a copy of the dense linear order $(\mathbb Q,\le)$. Motivated by a recent preprint of Richter, Stephan, and Zhang, which shows that bounded finite-one degrees can be as rigid as a discrete $ω$-chain and asks whether there are bounded finite-one degrees consisting exactly of a dense linearly ordered set of $1$-degrees, we introduce a block-density profile method for controlling one-one reducibility inside a single bounded finite-one degree. As further applications, in the same bounded finite-one degree we obtain an infinite antichain of $1$-degrees and, more generally, an embedded copy of every countable partial order. A single bounded finite-one degree can already exhibit dense, incomparable, and universal order-theoretic behaviour. Our main technical tool is a profile theorem based on computable block-density codings. The witness set constructed here is not $m$-rigid, so the phenomena obtained in this paper arise from a mechanism different from earlier $m$-rigidity-based constructions. Although our results do not settle the exact realization problem posed by Richter, Stephan, and Zhang, we show that density itself is not the obstruction: a single bounded finite-one degree may already contain a copy of $(\mathbb Q,\le)$, an infinite antichain, and embeddings of all countable partial orders.

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arXiv2026

A $wtt$-introimmune set in \texorpdfstring{$Π^0_1$}{Pi01} and introimmunity for several reducibilities

We prove that there exists a weak truth-table introimmune set in the class $Π^0_1$, settling the question left open in previous work of whether the known $Δ^0_2$ existence result can be improved to $Π^0_1$. Since $Σ^0_1$ sets cannot be immune, this is best possible for weak truth-table introimmunity. We also study introimmunity for Jockusch's bounded-search reducibility $\le_{bs}$ and Andersen's Dartmouth reducibility $\le_D$, proving the existence of $Δ^0_2$ sets that are $bs$-introimmune and $D$-introimmune; hence there also exists a $Δ^0_2$ $D^+$-introimmune set. We next consider the classical reducibility $\le_Q$, which is not contained in $\le_T$ on all subsets of $ω$. We show that no infinite $Π^0_1$ set is $Q$-introimmune, while a $Δ^0_2$ $Q$-introimmune set does exist. Thus the existence of $Δ^0_2$ $Q$-introimmune sets is best possible within the arithmetical hierarchy. Finally, for enumeration reducibility $\le_e$, we show that no infinite $Π^1_1$ set is $e$-introimmune, although $e$-introimmune sets do exist in the unrestricted sense. The proofs combine finite-injury priority arguments with dynamic spacing methods for $\le_{wtt}$, $\le_{bs}$, and $\le_D$, a bit-by-bit finite-extension construction for $\le_Q$, and an application of Soare's abstract existence theorem in the enumeration case.

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arXiv2026

$m$-Rigidity and Finite-One Degrees Inside Typical Many-One Degrees

In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one degrees inside many-one degrees, we study the finite-one structure of the many-one degree of an $m$-rigid set. First, combining bi-immunity of $m$-rigid sets with a theorem of Richter, Stephan, and Zhang, we show that for Lebesgue-almost every set $A$, and for a comeager class of sets $A$, the many-one degree $°_m(A)$ contains a least finite-one degree. Second, we prove that if $A$ is $m$-rigid, then $°_m(A)$ contains infinitely many pairwise incomparable finite-one degrees. More precisely, we construct representatives $B_S \equiv_m A$, indexed by computable sets $S$, such that $T \setminus S$ infinite implies $B_T \not\leq_{fin} B_S$. Third, inside a single finite-one degree we build a strict ascending chain \[ A_{(1)} <_1 A_{(2)} <_1 \cdots \] of $1$-degrees. These results yield almost-sure and comeager partial answers to the first two open problems posed by Richter, Stephan, and Zhang.

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arXiv2026

Rigid many-one degrees contain infinite antichains of $1$-degrees

Odifreddi asked whether every non-irreducible many-one degree must contain an infinite antichain of one-one degrees. Positive answers are known for computably enumerable many-one degrees (Degtev) and, more recently, for many-one degrees admitting a $Δ^0_2$ representative (Batyrshin). In this note we isolate a rigidity principle behind these phenomena. Call a set $A\subseteqω$ \emph{$m$-rigid} if every total computable $m$-autoreduction of $A$ is eventually the identity. We prove that if $A$ is $m$-rigid, then its many-one degree $°_m(A)$ contains an infinite antichain of $1$-degrees. The proof uses a uniform duplication construction: for each computable parameter $S$ we define $B_S\equiv_m A$ so that any injective reduction $B_S\le_1 B_T$ induces an $m$-autoreduction of $A$ and therefore forces $S\subseteq^{*}T$. Choosing an almost-inclusion infinite antichain of computable sets yields the desired infinite $1$-antichain inside $°_m(A)$. As applications, Jockusch's rigidity theorem implies that every $1$-generic set is $m$-rigid, giving a comeager family of positive instances. Moreover, $m$-rigidity holds with Lebesgue measure $1$ (indeed, every Martin-Löf random real is $m$-rigid). Consequently, Odifreddi's Question~5 has a positive answer \emph{with probability $1$} for a fair-coin random $A\in 2^ω$; any counterexample (if it exists) is confined to a null set (and, by genericity, also to a meager set).

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