Sandip Das, A. Lahiri, Soumen Nandi, Sagnik Sen · 5 authors
An ( n , m ) -graph is characterized by n types of arcs and m types of edges. A homomorphism of an ( n , m ) -graph G to an ( n , m ) -graph H , is a vertex mapping that preserves adjacency, direction, and type. The ( n , m ) -chromatic number of G , denoted by χ n , m ( G ) , is the minimum value of | V ( H ) | such that there exists a homomorphism of G to H . The theory of homomorphisms of ( n , m ) -graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic , Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of G is bounded by a function of χ n , m ( G ) but not the other way around. Additionally, we show that the acyclic chromatic number of G is bounded by a function of χ n , m ( G ) , a result already known in the reverse direction. Furthermore, we prove that the ( n , m ) -chromatic number for the family of graphs with maximum average degree less than 2 + 2 4 ( 2 n + m ) − 1 , including the subfamily of planar graphs with girth at least 8 ( 2 n + m ) , equals 2 ( 2 n + m ) + 1 . This improves upon previous findings, which proved the ( n , m ) -chromatic number for planar graphs with girth at least 10 ( 2 n + m ) − 4 is 2 ( 2 n + m ) + 1 . It is established that the ( n , m ) -chromatic number for the family T 2 of partial 2-trees is both bounded below and above by quadratic functions of ( 2 n + m ) , with the lower bound being tight when ( 2 n + m ) = 2 . We prove 14 ≤ χ ( 0 , 3 ) ( T 2 ) ≤ 15 and 14 ≤ χ ( 1 , 1 ) ( T 2 ) ≤ 21 which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.
Abstract This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K -algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t -designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -algebra $$\mathbb {Q}[V_n]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set $$[n]:=\{1,2,\ldots , n\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . All results are equally true if we replace $$\mathbb {Q}[V_n]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> by $$K[V_n]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , where K is any perfect field of characteristics $$\not =2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework.
This paper will give both the necessary and sufficient conditions required to find a counter-example to the Goldbach Conjecture by using an algebraic approach where no knowledge of the gaps between prime numbers is needed. To eliminate ambiguity the set of natural numbers, $\mathbb{N}$, will include zero throughout this paper. Also, for any sufficiently large $a \in \mathbb{N}$ the set $\mathcal{P}$ is the set of all primes $p_i \leq a$. It will be shown there exists a counter-example to the Goldbach Conjecture, given by $2a$ where $a \in \mathbb{N}_{> 3}$, if and only if for each prime $p_i \in \mathcal{P}$ there exists some unique $q_i, \alpha_i \in \mathbb{N}$ where $a 3$. However, this leads to contradiction since $2a 4$.A similar method will be employed to give the necessary and sufficient conditions when an even number is not the difference of two primes with one prime being less than that even number. To begin, let $a \in \mathbb{N}_{> 3}$ with the condition that the function $\gamma(a + 1)$ is equal to one if $a + 1$ is prime and zero otherwise. $2a$ is a counter-example if and only if for each prime $p_i \in \mathcal{P}$ there exists some unique $u_i, \beta_i \in \mathbb{N}$ where $2a 3$ to the equation above, leading to the same contradiction as the Goldbach Conjecture since $2a 4$. These proofs will have implications for proving the Polignac Conjecture.
Miha E. Habič, Joel David Hamkins, Lukas Daniel Klausner, Jonathan L. Verner · 5 authors
Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.
Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.
We prove that for every 3-player (3-prover) game G with value less than one, whose query distribution has the support S = {(1,0,0), (0,1,0), (0,0,1)} of Hamming weight one vectors, the value of the n-fold parallel repetition G^{⊗n} decays polynomially fast to zero; that is, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For every 3-player game G over binary questions and arbitrary answer lengths, with value less than 1, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}. Our proof technique is new and requires many new ideas. For example, we make use of the Level-k inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
Let Γ denote the modular group SL(2,Z) and Cn(Γ) the number of congruence subgproups of Γ of index at most n. We prove that lim n→∞ log Cn(Γ) (log n)2/ log log n = 3−2 √ 2 4 . Some extensions of this result for other arithmetic groups are presented as well as a general conjecture. §0. Introduction Let k be an algebraic number field, O its ring of integers, S a finite set of valuations of k (containing all the archimedean ones), and OS = { x ∈ k ∣∣ v(x) ≥ 0, ∀v ∈ S}. Let G be a semisimple, simply connected, connected algebraic group defined over k with a fixed embedding into GLd. Let Γ = G(OS) = G ∩ GLd(OS) be the corresponding S-arithmetic group. We assume that Γ is an infinite group. For every non-zero ideal I of OS let Γ(I) = Ker ( Γ → GLd(OS/I) ) . A subgroup of Γ is called a congruence subgroup if it contains Γ(I) for some I. For n > 0, define Cn(Γ) = # { congruence subgroups of Γ of index at most n } . Theorem 1. There exist two positive real numbers α− and α+ such that for all sufficiently large positive integers n n log n log log nα− ≤ Cn(Γ) ≤ n log n log log nα+ . This theorem is proved in [Lu], although the proof of the lower bound presented there requires the prime number theorem on arithmetic progressions in an interval where its validity depends on the GRH (generalized Riemann hypothesis for arithmetic progressions). The first two authors research is supported in part by the NSF. The third author’s Research is supported in part by OTKA T 034878. All three authors would like to thank Yale University for its hospitality. Typeset by AMS-TEX 1 2 DORIAN GOLDFELD ALEXANDER LUBOTZKY LASZLO PYBER In §2 below, we show that by appealing to a theorem of Linnik [Li1, Li2] on the least prime in an arithmetic progression, the proof can be made unconditional. Following [Lu] we define: α+(Γ) = lim logCn(Γ) λ(n) , α−(Γ) = lim logCn(Γ) λ(n) , where λ(n) = (log n) 2 log log n . It is not difficult to see that α+ and α− are independent of both the choice of the representation of G as a matrix group, as well as independent of the choice of S. Hence α± depend only on G and k. The question whether α+(Γ) = α−(Γ) and the challenge to evaluate them for Γ = SL2(Z) and other groups were presented in [Lu]. It was conjectured by Rademacher that there are only finitely many congruence subgroups of SL2(Z) of genus zero. This counting problem has a long history. Petersson [Pe, 1974] proved that the number of all subgroups of index n and fixed genus goes to infinity exponentially as n → ∞. Dennin [De, 1975] proved that there are only finitely many congruence subgroups of SL2(Z) of given fixed genus and solved Rademacher’s conjecture. It does not seem possible, however, to accurately count all congruence subgroups of index at most n in SL2(Z) by using the theory of Riemann surfaces of fixed genus. Here we prove: Theorem 2. α+(SL2(Z)) = α−(SL2(Z)) = 3−2 √ 2 4 = 0.0428932 . . . We believe that SL2(Z) represents the general case and we expect that α+ = α− for all groups. The proof of the lower bound in Theorem 2 is based on the Bombieri-Vinogradov Theorem [Bo], [Da], [Vi], i.e., the Riemann hypothesis on the average. The upper bound, on the other hand, is proved by first reducing the problem to a counting problem for subgroups of abelian groups and then solving that extremal counting problem. We will, in fact, show a more remarkable result: the answer is independent of O! Theorem 3. Let k be a number field with Galois group g = Gal(k/Q) and with ring of integers O. Let S be a finite set of primes, and OS as above. Assume GRH (generalized Riemann hypothesis) for k and all cyclotomic extensions k(ζ ) with a rational prime and ζ a primitive th root of unity. Then α+(SL2(OS)) = α−(SL2(OS)) = 3 − 2 √ 2 4 . The GRH is needed only for establishing the lower bound. It can be dropped in many cases by appealing to a theorem of Murty and Murty [MM] which generalizes the Bombieri– Vinogradov Theorem cited earlier. COUNTING CONGRUENCE SUBGROUPS 3 Theorem 4. Theorem 3 can be proved unconditionally for k if either (a) g = Gal(k/Q) has an abelian subgroup of index at most 4 (this is true, for example, if k is an abelian extension); (b) d = deg[k : Q] < 42. We conjecture that for every Chevalley group scheme G, the upper and lower limiting constants, α±(G(OS)), depend only on G and not on O. In fact, we have a precise conjecture, for which we need to introduce some additional notation. Let G be a Chevalley group scheme of dimension d = dim(G) and rank = rk(G). Let κ = |Φ+| denote the number of positive roots in the root system of G. Letting R = R(G) = d− 2 = κ , we see that R = +1 2 , (resp. , , −1, 3, 6, 6, 9, 15) if G is of type A (resp. B , C , D , G2, F4, E6, E7, E8). Conjecture. Let k,O, and S be as in Theorem 3, and suppose that G is a simple Chevalley group scheme. Then α+(G(OS)) = α−(G(OS)) = (√ R(R + 1) −R )2 4R2 . The conjecture reflects the belief that “most” subgroups of H = G(Z/mZ) lie between the Borel subgroup B of H and the unipotent radical of B. Our proof covers the case of SL2 and we are quite convinced that this will hold in general. For general G, we do not have such an in depth knowledge of the subgroups of G(Fq) as we do for G = SL2, yet we can still prove: Theorem 5. Let k,O, and S be as in Theorem 3. Let G be a simple Chevalley group scheme of dimension d and rank , and R = R(G) = d− 2 , then: (a) Assuming GRH or the assumptions of Theorem 4; α−(G(OS)) ≥ (√ R(R + 1) −R )2