Martin's Conjecture Research Articles

Topological reducibilities for discontinuous functions and their structures

In this article, we give a full description of the topological many-one degree structure of real-valued functions, recently introduced by Day—Downey—Westrick. We also clarify the relationship between the Martin conjecture and Day—Downey—Westrick’s topological Turing-like reducibility, also known as parallelized continuous strong Weihrauch reducibility, for single-valued functions: Under the axiom of determinacy, we show that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered; and moreover, if f has continuous Weihrauch rank α, then f′ has continuous Weihrauch rank α + 1, where f′(x) is defined as the Turing jump of f(x).

A duality of scaffolds for translation association schemes

Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished “root” nodes and with matrix edge weights, often taken from Bose–Mesner algebras. In this paper, we first present a slight modification of Martin's conjecture (2021) concerning a duality of scaffolds whose digraphs are embedded in a closed disk in the plane with root nodes all lying on the boundary circle, and then show that this modified conjecture holds true if we restrict ourselves to the class of translation association schemes, i.e., those association schemes that admit abelian regular automorphism groups.

Uniform Martin’s conjecture, locally

We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol x \le_T \boldsymbol y$. Besides improving our knowledge about part I of uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $\le_c$ on equivalence relations on $\mathbb N$ has a very complicated structure, as $\le_T$ is Borel reducible to it. We conclude raising the question Is part II of uniform Martin's conjecture implied by local phenomena, too? and briefly indicating a possible direction.

Turing degrees in Polish spaces and decomposability of Borel functions

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

Almost primes and the Banks–Martin conjecture

It has been known since Erdős that the sum of 1/(nlog⁡n) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k, and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k=6.

Martin's Conjecture: A Classification of the Naturally Occurring Turing Degrees

Martin's Conjecture: A Classification of the Naturally Occurring Turing Degrees

The uniform Martin’s conjecture for many-one degrees

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them a cone. We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^\omega$ and Wadge degrees of functions $\omega^\omega\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.

A family of optimal ternary cyclic codes from the Niho-type exponent

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. The objective of this paper is to present a family of optimal ternary cyclic codes from the Niho-type exponent. Specifically, for an odd integer m and a positive integer r with 4r≡1(modm), a family of cyclic codes C(u,v) of length 3m−1 over GF(3) with two nonzeros βu and βv is studied, where β is a generator of GF(3m)⁎, u=(3m+1)/2 and v=3r+2 is the ternary Niho-type exponent. The parameters of this family of cyclic codes are determined. It turns out that C(u,v) is optimal with respect to the Sphere Packing bound if 9∤m and otherwise almost optimal. Thanks to a recent proof of the Dobbertin–Helleseth–Kumar–Martin conjecture by Katz and Langevin, the dual of this family of cyclic codes is shown to have at most five nonzero weights.

The Liouville theorem under second order differentiability assumption

Iwaniec and Martin proved that in even dimensions n≥3, Wloc1,n/2 conformal mappings are Möbius transformations and they conjectured that it should also be true in odd dimensions. We prove this theorem for a conformal map f∈Wloc1,1 in dimension n≥3 under one additional assumption that the norm of the first order derivative |Df| satisfies |Df|p∈Wloc1,2 for p≥(n−2)/4. This is optimal in the sense that if |Df|p∈Wloc1,2 for p<(n−2)/4, it may not be a Möbius transform. This result shows the necessity of the Sobolev exponent in the Iwaniec–Martin conjecture. Meanwhile, we show that the Iwaniec–Martin conjecture can be reduced to a conjecture about the Caccioppoli type estimate.

On the degree-chromatic polynomial of a tree

The degree chromatic polynomial $P_m(G,k)$ of a graph $G$ counts the number of $k$ -colorings in which no vertex has m adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree. Le polynôme degré chromatique $P_m(G,k)$ d'un graphe $G$ compte le nombre de $k$-colorations dans lesquelles aucun sommet n'a m sommets adjacents de sa même couleur. On démontre la conjecture de Humpert et Martin sur les coefficients principaux du polynôme degré chromatique d'un arbre.

The strength of the projective Martin conjecture

We show that Martin’s conjecture on Π1-functions uniformly ≤T -order preserving on a cone implies Π1 Turing Determinacy over ZF+DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant Π2n+1functions is equivalent over ZFC to Σ2n+2-Axiom of Determinacy. As a corollary, the consistency of the conjecture for Π1-uniformly degree invariant functions implies the consistency of the existence of a Woodin cardinal.

Martin's conjecture holds for weight 3 blocks of symmetric groups

We prove that Martin's conjecture holds for weight 3 blocks of symmetric group algebras—that if these blocks have Abelian defect group, then their projective (indecomposable) modules have a common radical length 7.

A parametrised choice principle and Martin's conjecture on Blackwell determinacy

We define a parametrised choice principle PCP which (under the assumption of the Axiom of Blackwell Determinacy) is equivalent to the Axiom of Determinacy. PCP describes the difference between these two axioms and could serve as a means of proving Martin's conjecture on the equivalence of these axioms. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Kp,q-factorization of the complete bipartite graph Km,n

Let K m , n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A K p , q -factorization of K m , n is a set of edge-disjoint K p , q -factors of K m , n which partition the set of edges of K m , n . Martin (Discrete Math. 167/168 (1997) 461) gave simple necessary conditions for K p , q -factorization of K m , n , and conjectured these conditions are sufficient. In this paper, it is shown that Martin's conjecture on K p , q -factorization of K m , n is true for p : q =2:3.

The simulation technique and its applications to infinitary combinatorics under the axiom of Blackwell determinacy

The Axiom of Blackwell Determinacy is a set-theoretic axiom motivated by games used in statistics. It is known that the Axiom of Determinacy implies the Axiom of Blackwell Determinacy. Tony Martin has conjectured that the two axioms are equivalent. We develop the simulation technique which allows us to simulate boundedness proofs under the assumption of Blackwell Determinacy and deduce strong combinatorial consequences that can be seen as an important step towards proving Tony Martin's conjecture.

Kp, q-factorization of the complete bipartite graph Km, n

Let K m, n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A K p, q -factorization of K m, n is a set of edge-disjoint K p, q -factors of K m, n which partition the set of edges of K m, n . Martin (Discrete Math. 167/168 (1997) 461) gave simple necessary conditions for K p, q -factorization of K m, n , and conjectured these conditions are sufficient. In this paper, it is shown that Martin's conjecture on K p, q -factorization of K m, n is true for p: q=2:3.

Kp,q-factorization of complete bipartite graphs

Let K m , n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A K p , q -factorization of K m , n is a set of edge-disjoint K p , q -factors of K m , n which partition the set of edges of K m , n . When p = 1 and q is a prime number, Wang, in his paper “On K 1, k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K 1, q -factorization of K m , n and gave a sufficient condition for such a factorization to exist. In the paper “ K 1, k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang's result to the case that q is any positive integer. In this paper, we give a sufficient condition for K m , n to have a K p , q -factorization. As a special case, it is shown that the Martin's BAC conjecture is true when p : q = k :( k +1) for any positive integer k .

A remark on Martin's Conjecture

AbstractWe prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.

Modules with few types over a commutative valuation ring

It is proved that a module M over a countable commutative valuation ring has few models if and only if M is ω-stable. Let T be an arbitrary complete theory of modules over either a countable serial ring or a countable commutative Prufer ring or a countable hereditary Noetherian prime ring. It is proved that Martin's conjecture is true for T.

On a conjecture of Martin on the parameters of completely regular codes and the classification of the completely regular codes in the biggs-smith graph

In this paper we study a conjecture of Martin [7] on the paramaters of completely regular codes in distance-regular graphs. We show that this conjecture is not true in general, but for most classical graphs it is true. we also show that there is a counterexample in the Biggs-Smith graph for a weakened version of Martin's conjecture. Furthermore we classify the completely regular codes in the Biggs-Smith graph.