On Ramsey Minimal Graphs

10.37236/1184 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Tomasz Łuczak
Keyword(s):  
Infinite Number ◽  
Probabilistic Argument ◽  
Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest $F$ other than a matching, and every graph $G$ containing a cycle, there exists an infinite number of graphs $J$ such that $J\to (F,G)$ but if we delete from $J$ any edge $e$ the graph $J-e$ obtained in this way does not have this property.

scholarly journals On minimal Ramsey graphs and Ramsey equivalence in multiple colours

2020 ◽  
Vol 29 (4) ◽  
pp. 537-554
Author(s):  
Dennis Clemens ◽  
Anita Liebenau ◽  
Damian Reding
Keyword(s):  
Infinite Number ◽  
Maximum Degree ◽  
List Type ◽  
Minimal Graph ◽  
Induced Subgraph ◽  
Minimal Graphs ◽  
Large Maximum ◽  
Ramsey Minimal Graph ◽  
Ramsey Graphs ◽  
Monochromatic Copy

AbstractFor an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences.For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H.For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number.The collection $\{\mathcal M_q(H) \colon H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where $\mathcal M_q(H)$ denotes the set of all graphs that are q-Ramsey-minimal for H.We also address the question of which pairs of graphs satisfy $\mathcal M_q(H_1)=\mathcal M_q(H_2)$ , in which case H1 and H2 are called q-equivalent. We show that two graphs H1 and H2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

What is the purpose of cognition?

2020 ◽  
Vol 43 ◽  
Author(s):  
Aba Szollosi ◽  
Ben R. Newell
Keyword(s):  
Infinite Number ◽  
Human Cognition ◽  
The Way

Abstract The purpose of human cognition depends on the problem people try to solve. Defining the purpose is difficult, because people seem capable of representing problems in an infinite number of ways. The way in which the function of cognition develops needs to be central to our theories.

The Problem of Alternative Monotheisms: Another Serious Challenge to Theism

2018 ◽  
Vol 10 (1) ◽  
pp. 31-51
Author(s):  
Raphael Lataster
Keyword(s):  
Infinite Number ◽  
Current Evidence ◽  
God Concept ◽  
Classical Theism ◽  
The World ◽  
Divine Transcendence ◽  
Hiddenness Of God

Theistic and analytic philosophers of religion typically privilege classical theism by ignoring or underestimating the great threat of alternative monotheisms.[1] In this article we discuss numerous god-models, such as those involving weak, stupid, evil, morally indifferent, and non-revelatory gods. We find that theistic philosophers have not successfully eliminated these and other possibilities, or argued for their relative improbability. In fact, based on current evidence – especially concerning the hiddenness of God and the gratuitous evils in the world – many of these hypotheses appear to be more probable than theism. Also considering the – arguably infinite – number of alternative monotheisms, the inescapable conclusion is that theism is a very improbable god-concept, even when it is assumed that one and only one transcendent god exists.[1] I take ‘theism’ to mean ‘classical theism’, which is but one of many possible monotheisms. Avoiding much of the discussion around classical theism, I wish to focus on the challenges in arguing for theism over monotheistic alternatives. I consider theism and alternative monotheisms as entailing the notion of divine transcendence.

Living Mirrors

2019 ◽  
Author(s):  
Ohad Nachtomy
Keyword(s):  
Infinite Number ◽  
Historical Background ◽  
The Absolute

This work presents Leibniz’s view of infinity and the central role it plays in his theory of living beings. Chapter 1 introduces Leibniz’s approach to infinity by presenting the central concepts he employs; chapter 2 presents the historical background through Leibniz’s encounters with Galileo and Descartes, exposing a tension between the notions of an infinite number and an infinite being; chapter 3 argues that Leibniz’s solution to this tension, developed through his encounter with Spinoza (ca. 1676), consists of distinguishing between a quantitative and a nonquantitative use of infinity, and an intermediate degree of infinity—a maximum in its kind, which sheds light on Leibniz’s use of infinity as a defining mark of living beings; chapter 4 examines the connection between infinity and unity; chapter 5 presents the development of Leibniz’s views on infinity and life; chapter 6 explores Leibniz’s distinction between artificial and natural machines; chapter 7 focuses on Leibniz’s image of a living mirror, contrasting it with Pascal’s image of a mite; chapter 8 argues that Leibniz understands creatures as infinite and limited, or as infinite in their own kind, in distinction from the absolute infinity of God; chapter 9 argues that Leibniz’s concept of a monad holds at every level of reality; chapter 10 compares Leibniz’s use of life and primitive force. The conclusion presents Leibniz’s program of infusing life into every aspect of nature as an attempt to re-enchant a view of nature left disenchanted by Descartes and Spinoza.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

scholarly journals Higher spins from exotic dualisations

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Nicolas Boulanger ◽  
Victor Lekeu
Keyword(s):  
Infinite Number ◽  
Arbitrary Number ◽  
Degrees Of Freedom ◽  
Young Tableaux ◽  
Spacetime Dimension ◽  
Massless Field ◽  
Free Level ◽  
Higher Spins

Abstract At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.

scholarly journals A brief review of E theory

2016 ◽  
Vol 31 (26) ◽  
pp. 1630043 ◽  
Author(s):  
Peter West
Keyword(s):  
Infinite Number ◽  
Equations Of Motion ◽  
Space Time ◽  
Unified Theory ◽  
Dynamical Equations ◽  
Supergravity Theory ◽  
Maximal Supergravity ◽  
Abdus Salam ◽  
Strings And Branes ◽  
Time Lead

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra [Formula: see text] and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of [Formula: see text] we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the [Formula: see text] conjecture given many years ago.

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