Pair Of Disjoint Sets Research Articles

On the Ramsey numbers of daisies I

Abstract Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An $r$ -daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$ -uniform hypergraph $\{K\cup P\,{:}\, P\in M^{(r)}\}$ . Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$ -daisies and also for special cases where the size of the kernel is bounded.

Generalized Frame for Orthopair Fuzzy Sets: (m,n)-Fuzzy Sets and Their Applications to Multi-Criteria Decision-Making Methods

Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “(m,n)-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank (m,n)-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with (m,n)-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of (m,n)-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example.

Distinguishing Vagueness from Ambiguity in Rough Set Approximations

In this paper we present a new approach to rough set approximations that permits to distinguish between two kinds of “imperfect” knowledge in a joint framework: on one hand, vagueness, due to imprecise knowledge and uncertainty typical of fuzzy sets, and on the other hand, ambiguity, due to granularity of knowledge originating from the coarseness typical of rough sets. The basic idea of our approach is that each concept is represented by an orthopair, that is, a pair of disjoint sets in the universe of knowledge. The first set in the pair contains all the objects that are considered as surely belonging to the concept, while the second set contains all the objects that surely do not belong to the concept. In this context, following some previous research conducted by us on the algebra of rough sets, we propose to define as rough approximation of the orthopair representing the considered concept another orthopair composed of lower approximations of the two sets in the first orthopair. We shall apply this idea to the classical rough set approach based on indiscernibility, as well as to the dominance-based rough set approach. We discuss also a variable precision rough approximation, and a fuzzy rough approximation of the orthopairs. Some didactic examples illustrate the proposed methodology.

Vertex-minors and the Erdős–Hajnal conjecture

We prove that for every graph H, there exists ε>0 such that every n-vertex graph with no vertex-minors isomorphic to H has a pair of disjoint sets A, B of vertices such that |A|,|B|≥εn and A is complete or anticomplete to B. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erdős–Hajnal conjecture for vertex-minors.

Colourings without monochromatic disjoint pairs

The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erdős–Rothschild problem, introduced in 1974 by Erdős and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure.The celebrated Erdős–Ko–Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erdős–Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces.In this paper we present a unified approach to the Erdős–Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erdős–Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann.

Orthopairs and granular computing

Pairs of disjoint sets (orthopairs) naturally arise or have points in common with many tools to manage uncertainty: rough sets, shadowed sets, version spaces, three-valued logics, etc. Indeed, they can be used to model partial knowledge, borderline cases, consensus, examples and counter-examples pairs. Moreover, generalized versions of orthopairs are the well known theories of Atanassov intuitionistic fuzzy sets and possibility theory and the newly established three-way decision theory. Thus, it is worth studying them on an abstract level in order to outline general properties that can then be casted to the different paradigms they are in connection with. In this paper, we will review how to define orthopairs and a hierarchy on them in the light of granular computing. Aggregation operators will also be discussed as well as possible generalizations and connections with different paradigms. This will permit us to point out new facets of these paradigms and outline some possible future developments.

A Probability Measure for Partial Events

We introduce the concept of partial event as a pair of disjoint sets, respectively the favorable and the unfavorable cases. Partial events can be seen as a De Morgan algebra with a single fixed point for the complement. We introduce the concept of a measure of partial probability, based on a set of axioms resembling Kolmogoroff’s. Finally we define a concept of conditional probability for partial events and apply this concept to the analysis of the two-slit experiment in quantum mechanics.

Remarks about disjoint dominating sets

We prove several structural and hardness results concerning pairs of disjoint sets in graphs which are dominating or independent and dominating.

Measuring Resemblances Between Swarm Behaviours: A Perceptual Tolerance Near Set Approach

The problem considered in this article is how to detect and measure resemblances between swarm behaviours. The solution to this problem stems from an extension of recent work on tolerance near sets and image correspondence. Instead of considering feature extraction from subimages in digital images, we compare swarm behaviours by considering feature extraction from subsets of tuples of feature-values representing the behaviour of observed swarms of organisms. Thanks to recent work on the foundations of near sets, it is possible to formulate a rigorous approach to measuring the extent that swarm behaviours resemble each other. Fundamental to this approach is what is known as a recent description-based set intersection, a set containing objects with matching or almost the same descriptions extracted from objects contained in pairs of disjoint sets. Implicit in this work is a new approach to comparing information tables representing N. Tinbergen's ethology (study of animal behaviour) and direct result of recent work on what is known as rough ethology. Included in this article is a comparison of recent nearness measures that includes a new form of F. Hausdorff's distance measure. The contribution of this article is a tolerance near set approach to measuring the degree of resemblance between swarm behaviours.

Minimal paths and cycles in set systems

A minimal k -cycle is a family of sets A 0 , … , A k − 1 for which A i ∩ A j ≠ 0̸ if and only if i = j or i and j are consecutive modulo k . Let f r ( n , k ) be the maximum size of a family of r -sets of an n element set containing no minimal k -cycle. Our results imply that for fixed r , k ≥ 3 , ℓ n − 1 r − 1 + O ( n r − 2 ) ≤ f r ( n , k ) ≤ 3 ℓ n − 1 r − 1 + O ( n r − 2 ) , where ℓ = ⌊ ( k − 1 ) / 2 ⌋ . We also prove that f r ( n , 4 ) = ( 1 + o ( 1 ) ) n − 1 r − 1 as n → ∞ . This supports a conjecture of Z. Füredi [Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (2–3) (1984) 161–168] on families in which no two pairs of disjoint sets have the same union.