We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinalμ, of cofinalityω, such that everyμ+-chromatic graphXonμ+has an edge colouringcofXintoμcolours for which every vertex colouringgofXinto at mostμmany colours has ag-colour class on whichctakes every value.The paper also contains some generalisations of the above statement in whichμ+is replaced by other cardinals >μ.

We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$ , of cofinality $\omega$ , such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ many colours has a g-colour class on which c takes every value. The paper also contains some generalisations of the above statement in which $\mu^{+}$ is (...) replaced by other cardinals < $\mu$. (shrink)

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and (...) culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica26(2) (2006) 183–205], the Ramsey theory of H3 had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin.2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math.185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph G, there is a finite number T(G) such that for any coloring of all copies of G in H3 into finitely many colors, there is a subgraph of H3 which is again universal homogeneous triangle-free in which the coloring takes no more than T(G) colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code H3 and the development of their Ramsey theory. (shrink)

Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author’s recent result for the triangle-free Henson graph, we prove (...) that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey’s Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker. (shrink)

We observe a number of connections between recent developments in the study of constraint satisfaction problems, irredundant axiomatisation and the study of topological quasivarieties. Several restricted forms of a conjecture of Clark, Davey, Jackson and Pitkethly are solved: for example we show that if, for a finite relational structure M, the class of M-colourable structures has no finite axiomatisation in first order logic, then there is no set (even infinite) of first order sentences characterising the continuously M-colourable structures amongst compact (...) totally disconnected relational structures. We also refute a rather old conjecture of Gorbunov by presenting a finite structure with an infinite irredundant quasi-identity basis. (shrink)

We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property, We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. (...) As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particlar, we apply this technique to graph colourings and a class of partial algebras arising from separation logic. (shrink)

Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s $\textsf{ZFC}$ bound on $2^{\aleph _\omega }$. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jónsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many (...) more.In this paper, the first part of a series, we survey various forms of club guessing that have appeared in the literature, and then systematically study the various ways in which a club guessing sequence can be improved, especially in the way the frequency of guessing is calibrated.We include an expository section intended for those unfamiliar with club guessing and which can be read independently of the rest of the article. (shrink)

We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the least number of (...) colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs. (shrink)

The squared paper or graphs are grid-based design representations used in engineering, industrial and craft design practices wherein designs are drawn over symmetrical grids. This paper reports grid-processing strategies undertaken by actors in a native craft practice, viz. Kashmiri carpet-weaving having three task contexts: (1) _design_, wherein designs are drawn on graph sheets and color scheme given by assigning practice-specific symbolic codes to the motifs by designers; (2) _coding_, wherein a cryptic script, called _talim_, is generated from these encoded (...) graphs by talim-writers; and (3) _weaving_, wherein this script is read, decoded and communicated in an equally cryptic trade-language by weavers among their teams. 32 video-recorded think-aloud sessions were conducted with talim-writers to understand their graph-processing and resultant coding strategies. The observations and qualitative analysis of transcripts revealed highly _situated_ strategies undertaken by experts keen on imbibing visuality of design embedded in the grids, in contrast to _embodied_ strategies undertaken by less-experienced coders keen on processing grids’ structural features and getting overwhelmed by grid-clutter in the process. In this landscape, the findings revealed a negotiated nature of errors instead of being deviations from the underlying grid and thereby objectively assessable. The paper makes recommendations for CAD systems to include more of those grid features exploited by experts during their graph processing to facilitate designers. (shrink)

Our conscious experiences are qualitative and unitary. The qualitative universals given in particular experiences, i.e. qualia, combine into the seamless unity of our conscious experience. The problematics of quality and cohesion are not unique to consciousness studies. In mathematics, the study of qualities (e.g., shape) resulting from quantitative variations in cohesive spaces led to the axiomatization of cohesion and quality. Using the mathematical definition of quality, herein we model qualia space as a categorical product of qualities. Thus modeled qualia space (...) is a codomain space wherein composite qualities (e.g. shape AND color) of conscious experiences can be valued. As part of characterizing the qualia space, we provide a detailed exemplification of the mathematics of quality and cohesion in terms of the categories of idempotents and reflexive graphs. More specifically, with qualities as commutative triangles formed of cohesion-preserving functors, first we calculate the product of commutative triangles. Next, we explicitly show that the category of idempotents is a quality type. Lastly, as part of showing that the category of reflexive graphs is cohesive, we characterize the adjointness between functors relating cohesive graphs to discrete sets. In conclusion, our category theoretic construction of qualia space is a formalization of the binding of qualitative features (colors and shapes) into the cohesive objects (colored-shapes) of conscious experiences. Compared to the feature-vector accounts of conscious experiences, our product-of-qualities account of consciousness is a substantial theoretical advance. (shrink)

We consider modal logics whose intermediate fragments lie between the logic of infinite problems [20] and the Medvedev logic of finite problems [15]. There is continuum of such logics [19]. We prove that none of them is finitely axiomatizable. The proof is based on methods from [12] and makes use of some graph-theoretic constructions (operations on coverings, and colourings).

Maps and realist pictures comprise prominent sub-classes of iconic representations. The most basic, most important sub-class is perception. Other types are drawings, photographs, musical notations, diagrams, bar graphs, abacuses, hieroglyphs, and color chits.

In early June, 2020, Fox News displayed for viewers a color-coded graph that compared the amounts that the stock market had risen in the aftermath of major racial tragedies, including the assassina...

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Rembrandt BookProfessor Emeritus John Adkins RichardsonThe Rembrandt Book by Gary Schwartz. New York: Harry N. Abrams, 2006, 384 pp. $40.95, cloth.This truly is the Rembrandt book. Substantial in every way, it is physically imposing, magnificently printed on heavy, glossy stock and profusely illustrated with splendid color reproductions of all the master’s major works and many sketches and preparatory drawings, as well as etchings and dry-point engravings. Gary (...) Schwartz, who studied art history at New York University and Johns Hopkins University, is now living in the Netherlands. The merits of his book are enormous. Not only is it filled with magnificent reproductions, there are tables and graphs detailing such things as the number of Rembrandt paintings in catalogues from 1836 to 1992; analyses of the artist’s borrowings and adaptations in five-year periods; key dates in the life of the man; Rembrandt’s relation to the forms of sale for paintings in the Dutch seventeenth century; and so on. There is even a computer rendering of the famous depiction of Captain Banning Cocq’s Company of Guards at the Gates of Amsterdam (popularly called The Night Watch), specifying the weapons shown and identifying all present, either by name and title or by a description, such as “girl in gold and blue,” “pikeman,” “musketeer,” etc.A sampling of the chapter headings and subcategories suggests just how comprehensive is Schwartz’s treatment: 1. Introduction, 2. A Dutch Artist’s Life, 3. Family, Loved Ones, Households, 4. Craft, 5. Earning and Spending, 6. Patrons, 7. Landscape, 8. Humankind, 9. Man and God. Just the specifics of the last chapter are remarkably capacious: Holy Family; Christ’s Ministry; Passion, Death and Afterlife; Divine Intervention; Holy People; Moral Examples from the Old Testament; The Man of God, Simeon; Afterlife; and Rembrandt as a Christian Rhetorician of the Brush. It’s hard to imagine a better reference for the lay reader than The Rembrandt Book.The book is also full of interesting contrasts of opinion as well as some surprising remarks. As for one of the latter, Schwartz writes that Rembrandt’s “portraits are said to be more ‘psychological’ than those of his contemporaries. What this might mean I frankly do not understand” (213). He goes on to note that the field of psychology did not exist in Rembrandt’s time. However, from long before its invention people spoke of Rembrandt’s ability to reveal the soul of the sitter. Indeed, the reason I was not surprised, although somewhat distressed, when the Rembrandt Research Project—a group of connoisseurs set up in 1969 with the explicit charge of authenticating all works ascribed to the master—announced in 1985 that The Man with a Golden Helmet in Berlin was not by the master at all, was that I had all along been troubled by this face that lacked the quality of animation in repose so [End Page 115] characteristic of Rembrandt portraits. Schwartz thinks the artist would have to have relied upon the ancient theory of the humors to do “characterological descriptions” of his sitters, and that in portraiture, “such characterizations would quickly resemble caricatures.” And, indeed, they do, but in profoundly subtle ways, much as someone like the popular illustrator Norman Rockwell employed tiny caricature-like distortions to project a kind of homely Americanism. I’m not comparing Rockwell to Rembrandt but choosing a realistic illustrator in whose work the trick is more apparent. Caricature can be as broad as it is in Jacques Callot, William Hogarth, or Honoré Daumier and yet be a feature of true art. Or so it seems to me, admittedly, a sometimes cartoonist as well as painter and engraver.Schwartz’s completeness of coverage leads him to include, in addition to superb drawings and prints, some that are so careless and hasty as to make one doubt their authenticity. One of the works of marginal quality is an etching of Six’s Bridge, which is said to have been created outdoors, hastily, on the site. This is a hotly debated issue. Rembrandt is known to have carried “varnished”—that is, grounded—copper etching plates, so there wouldn’t seem much reason to... (shrink)

We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).

I work every day in the Cornell College Library. Usually on the ground floor level, where the fast computers are. The other day I took an early afternoon break, and went up to the second floor reading room to get a copy of The Times and relax. As I passed through the reading room I saw a Japanese student sitting at the long reading table, studying his physics text. He was sitting up straight; the hard back book was standing vertical (...) in front of him on the table, three feet away. The room was pretty silent and the boy was concentrating. I got my paper and took it back to the lower level to read. I read for twenty minutes, then returned to my computer, where I worked for an hour and a half. I finished my online teaching for the day, I packed up my stuff, climbed back up the stairs to the reading room, crossed the room to exit. At that point, tired and stretching, I usually don’t observe much. But this time I noticed the same Japanese student. He was still in the same position, sitting upright looking at his book, balancing it firmly in both hands. He seemed still to be in the same position of attention: nothing had changed, I felt, except the design on the page: a different graph of a different color. I went out through the main door of the Library, and went home. (shrink)

Several papers (e.g. [7, 23, 42]) have recently sought to give general frameworks for online structures and algorithms ([4]), and seeking to connect, if only by analogy, online and computable structure theory. These initiatives build on earlier work on online colouring and other combinatorial algorithms by Bean [10], Kierstead, Trotter et al. [48, 54, 57] and others, as we discuss below. In this paper we will look at such frameworks and illustrate them with examples from the first author’s MSc Thesis (...) ([58]). In this thesis, Askes looks at online, adversarial online and strongly online algorithms and structures. Additionally, we prove some new theorems about online algorithms on graphs of bounded pathwidth as well as some new results on punctually 1-decidable structures. (shrink)

With the rapid development of artificial intelligence technology, multitasking textual translation has attracted more and more attention. Especially after the application of deep learning technology, the performance of multitask translation text detection and recognition has been greatly improved. However, because multitasking contains the interference problem faced by the translated text, there is a big gap between recognition performance and actual application requirements. Aiming at multitasking and translation text detection, this paper proposes a text localization method based on multichannel multiscale detection (...) of the largest stable extreme value region and cascade filtering. This paper selects the appropriate color channel and scale to extract the maximum stable extreme value area as the character candidate area and designs a cascaded filter from coarse to fine to remove false detections. The coarse filter is based on some simple morphological features and stroke width features, and the fine filter is trained by a two-recognition convolutional neural network. The remaining character candidate regions are merged into horizontal or multidirectional character strings through the graph model. The experimental results on the text data set prove the effectiveness of the improved deep learning network character model and the feasibility of the textual implication translation analysis method based on this model. Among them, the text contains translation character recognition results prove that the model has good description ability. The characteristics of the model determine that this method is not sensitive to the scale of the sliding window, so it performs better than the existing typical methods in retrieval tasks. (shrink)

When describing variation at the lexical level in sign languages, researchers often distinguish between phonological and lexical variants, using the following principle: if two signs differ in only one of the major phonological components, then they are considered phonological variants, otherwise they are considered separate lexemes. We demonstrate that this principle leads to contradictions in some simple and more complex cases of variation. We argue that it is useful to visualize the relations between variants as graphs, and we describe possible (...) networks of variants that can arise using this visualization tool. We further demonstrate that these scenarios in fact arise in the case of variation in color terms and kinship terms in Russian Sign Language, using a newly created database of lexical variation in RSL. We show that it is possible to develop a set of formal rules that can help distinguish phonological and lexical variation also in the problematic scenarios. However, we argue that it might be a mistake to dismiss the actual patterns of variant relations in order to arrive at the binary lexical vs. phonological variant opposition. (shrink)

Paraconsistent logic is an area of philosophical logic that has yet to find acceptance from a wider audience. The area remains, in a word, disreputable. In this essay, we try to reassure potential consumers that it is not necessary to become a radical in order to use paraconsistent logic. According to the radicals, the problem is the absurd classical account of contradiction: Classically inconsistent sets explode only because bourgeois classical semantics holds, in the face of overwhelming evidence to the contrary, (...) that both A and ∼ A cannot simultaneously be true! We suggest (more modestly) that there is, at least sometimes, something else worth preserving, even in an inconsistent, unsatisfiable premise set. In this paper we present, in a new guise, a very general version of this “preservationist” approach to paraconsistency. (shrink)

A speaker can express the same thought, true under the same conditions, while using different expressions and grammatical constructions. According to Frege, these are differences in colourings. Colourings may convey additional contents; in that, they resemble Gricean conventional implicatures. Sander (2019) argues that Gricean implicatures do not subsume the category of colourings, as some colourings do not communicate their content. I show that this argument relies on a notion of communication focused on the speaker's intentions. But (...) a notion of communicative intentions where a speaker is responsible for the intentions her audience ascribes her possible. Under this notion, since so-called non-communicative colourings trigger specific inferences, a speaker who uses them communicates these inferences. Therefore, I vindicate the communicative role of colourings with content. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

We recall and improve our 2019 and 2020 concepts of n-SuperHyperGraph, Plithogenic nSuperHyperGraph, n-Power Set of a Set, and we present some application from the real world. The nSuperHyperGraph is the most general form of graph today and it is able to describe the complex reality we live in, by using n-SuperVertices (groups of groups of groups etc.) and nSuperHyperEdges (edges connecting groups of groups of groups etc.).

Christopher Meek. Related Graphical Frameworks: Undircted, Directed Acyclic and Chain Graph Models.

A new approach to policy analysis is formulated within the framework of the graph model for conflict resolution. A policy is defined as a plan of action for a decision maker (DM) that specifies the DM’s intended action starting at every possible state in a graph model of a conflict. Given a profile of policies, a Policy Stable State (PSS) is a state that no DM moves away from (according to its policy), and such that no DM would (...) prefer to change its policy given the policies of the other DMs. The profile of policies associated to a PSS is called a Policy Equilibrium. Properties of PSSs are developed, and a refinement is suggested that restricts DMs to policies that are credible in that they are in the DM’s immediate interest. Relationships with existing stability definitions in the graph model for conflict resolution are then explored. (shrink)

We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.

In scientific work, sonification is primarily thought of as a novel way to communicate post hoc research findings to lay audiences but only rarely, if ever, as a component of the research itself. This article argues that, rather than any inherent epistemological limitations of sound as a medium of scientific reasoning, this framing reflects a sociohistorical tendency to “silence” experimental techniques as they become widely adopted—both in terms of the literal silencing of noisy instrumentation and the elision of the role (...) of sound in received discovery accounts. This tendency is well-illustrated by the history of electrophysiology research, particularly the case of the electrical penetration graph, an instrument used by entomologists to study how insects feed on plants. Combining participant observation, interviews, and analysis of EPG technical literature, this article recovers the unacknowledged role of sound in EPG’s exploratory stages and gradual emergence as a standard laboratory technique as well as its gradual silencing as a consequence of this standardizing process. (shrink)

Slagging-off is an important preprocessing operation of steel-making to improve the purity of iron. Current manual-operated slag removal schemes are inefficient and labor-intensive. Automatic slagging-off is desirable but challenging as the reliable recognition of iron and slag is difficult. This work focuses on realizing an efficient and accurate recognition algorithm of iron and slag, which is conducive to realize automatic slagging-off operation. Motivated by the recent success of deep learning techniques in smart manufacturing, we introduce deep learning methods to this (...) field for the first time. The monotonous gray value of industry images, poor image quality, and nonrigid feature of iron and slag challenge the existing fully convolutional networks. To this end, we propose a novel spatial and feature graph convolutional network module. SFGCN module can be easily inserted in FCNs to improve the reasoning ability of global contextual information, which is helpful to enhance the segmentation accuracy of small objects and isolated areas. To verify the validity of the SFGCN module, we create an industrial dataset and conduct extensive experiments. Finally, the results show that our SFGCN module brings a consistent performance boost for a wide range of FCNs. Moreover, by adopting a lightweight network as backbone, our method achieves real-time iron and slag segmentation. In the future work, we will dedicate our efforts to the weakly supervised learning for quick annotation of big data stream to improve the generalization ability of current models. (shrink)

This study employs a knowledge graph approach to realize the representation and association of information resources, promote the research, teaching, and dissemination of Dunhuang cultural heritage (CH). The Dunhuang Mogao Grottoes is a UNESCO world CH site, and digitization of Dunhuang CH has produced a large amount of information resources. However, these digitized resources continue to lack the systematic granular semantic representation required to correlate Dunhuang cultural heritage information (CHI) in order to facilitate efficient research and appreciation. To respond (...) to this need, new approaches for representing CHI are being developed. This study identifies five facets and their semantic relationship to Dunhuang CH, constructs an ontology model to regulate the entities, attributes, and relationships of Dunhuang CH knowledge, and subsequently processes the resulting data using various techniques (such as semantic annotation and entity association) to facilitate rendering the data in a knowledge graph construction. Finally, we constructed a DH-oriented knowledge graph service platform in order to provide a user friendly visual display and semantic retrieval service. (shrink)