Information: - In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not. - The Zarankiewicz problem , an unsolved problem in mathematics , asks for the largest possible number of edges in a bipartite graph that has a given number of vertices but has no complete bipartite subgraphs of a given size . It belongs to the field of extremal graph theory , a branch of combinatorics , and is named after the Polish mathematician Kazimierz Zarankiewicz , who proposed several special cases of the problem in 1951 . The Kvari -- Sos -- Turan theorem , named after Tamas Kvari , Vera T. Sos , and Pal Turan , provides an upper bound on the solution to the Zarankiewicz problem . When the forbidden complete bipartite subgraph has one side with at most three vertices , this bound has been proven to be within a constant factor of the correct answer . For larger forbidden subgraphs , it remains the best known bound , and has been conjectured to be tight . Applications of the Kvari -- Sos -- Turan theorem include bounding the number of incidences between different types of geometric object in discrete geometry . - In mathematics, topology (from the Greek , "place", and , "study") is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness. - Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics). - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. - Extremal graph theory is a branch of the mathematical field of graph theory. Extremal graph theory studies extremal (maximal or minimal) graphs which satisfy a certain property. Extremality can be taken with respect to different graph invariants, such as order, size or girth. More abstractly, it studies how global properties of a graph influence local substructures of the graph. - In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets formula_1 and formula_2 (that is, formula_1 and formula_2 are each independent sets) such that every edge connects a vertex in formula_1 to one in formula_2. Vertex sets formula_1 and formula_2 are usually called the "parts" of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. - Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. - Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets "S" indexed by the natural numbers, enumerative combinatorics seeks to describe a "counting function" which counts the number of objects in "S" for each "n". Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. - Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of "balance" and/or "symmetry". These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in Sudoku grids. - In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not feasible. It operates on the domain of those optimization problems, in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Some common problems involving combinatorial optimization are the travelling salesman problem ("TSP") and the minimum spanning tree problem ("MST"). - Kazimierz Zarankiewicz (2 May 1902 5 September 1959) was a Polish mathematician, interested primarily in topology. - Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. - In mathematics graph theory is the study of "graphs", which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of "vertices", "nodes", or "points" which are connected by "edges", "arcs", or "lines". A graph may be "undirected", meaning that there is no distinction between the two vertices associated with each edge, or its edges may be "directed" from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics. - In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions. After reading the paragraphs above, we are interested in knowing the entity with which 'zarankiewicz problem' exhibits the relationship of 'instance of'. Find the answer from the choices below. Choices: - 1 - algebra - area - area of mathematics - bipartite graph - branch - change - collection - computer - design - economics - engineering - field - function - goal - graph - graph theory - group - intersection - mathematical problem - mathematics - maximal - may - medicine - model - number - object - part - problem - quantity - rank - science - size - space - structure - study - theory - tree - vector
The answer to this question is:
mathematical problem