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Counting Lemma
The counting lemmas this article discusses are statements in combinatorics and graph theory. The first one extracts information from \epsilon-regular pairs of subsets of vertices in a graph G, in order to guarantee patterns in the entire graph; more explicitly, these patterns correspond to the count of copies of a certain graph H in G. The second counting lemma provides a similar yet more general notion on the space of graphons, in which a scalar of the cut distance between two graphs is correlated to the homomorphism density between them and H. Graph embedding version of counting lemma Whenever we have an \epsilon-regular pair of subsets of vertices U,V in a graph G, we can interpret this in the following way: the bipartite graph, (U,V), which has density d(U,V), is ''close'' ''to being'' a random bipartite graph in which every edge appears with probability d(U,V), with some \epsilon error. In a setting where we have several clusters of vertices, some of the pairs between these c ...
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics i ...
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Graph Theory
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'' (also called ''nodes'' or ''points'') which are connected by ''edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ...
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Homomorphism Density
In the mathematical field of extremal graph theory, homomorphism density with respect to a graph H is a parameter t(H,-) that is associated to each graph G in the following manner: : t(H,G):=\frac. Above, \operatorname(H,G) is the set of graph homomorphisms, or adjacency preserving maps, from H to G. Density can also be interpreted as the probability that a map from the vertices of H to the vertices of G chosen uniformly at random is a graph homomorphism. There is a connection between homomorphism densities and subgraph densities, which is elaborated on below. Examples * The edge density of a graph G is given by t(K_,G). * The number of walks with k-1 steps is given by \operatorname(P_k, G). *\operatorname(C_k, G) = \operatorname(A^k) where A is the adjacency matrix of G. *The proportion of colorings using k colors that are proper is given by t(G, K_k). Other important properties such as the number of stable sets or the maximum cut can be expressed or estimated in terms of h ...
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Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denotin ...
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Graphon
GraphOn GO-Global is a multi-user remote access application for Windows. Overview GO-Global allows multiple users to concurrently run Microsoft Windows applications installed on a Windows server or server farm  from network-connected locations and devices. GO-Global redirects the user interface of Windows applications running on the Windows server to the display or browser on the user's device. Applications look and feel like they are running on the user's device. Supported end-user devices include Windows, Mac and Linux personal computers, iOS and Android mobile devices, and Chromebooks. GO-Global is used by Independent Software Vendors (ISVs), Hosted Service Providers (HSPs), and Managed Service Providers (MSPs) to publish Windows applications without modification of existing code for the use of local and remote users. Architecture GO-Global enables multi-user remote access to Windows applications without the use of Microsoft Remote Desktop Services (RDS) or th ...
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Graph Removal Lemma
In graph theory, the graph removal lemma states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. The special case in which the subgraph is a triangle is known as the triangle removal lemma. The graph removal lemma can be used to prove Roth's theorem on 3-term arithmetic progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem. It also has applications to property testing. Formulation Let H be a graph with h vertices. The graph removal lemma states that for any \epsilon > 0, there exists a constant \delta = \delta(\epsilon, H) > 0 such that for any n-vertex graph G with fewer than \delta n^h subgraphs isomorphic to H, it is possible to eliminate all copies of H by removing at most \epsilon n^2 edges from G. An alternative way to state this is to say that for any n-vertex graph G with o(n^h) subgraphs isomorphic to H, it is possible to el ...
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Lemmas
Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a part of a grass plant * Lemma (mathematics), a type of proposition Other uses * ''Lemma'' (album), by John Zorn (2013) * Lemma (logic), an informal contention See also *Analemma, a diagram showing the variation of the position of the Sun in the sky *Dilemma *Lema (other) * Lemmatisation Lemmatisation ( or lemmatization) in linguistics is the process of grouping together the inflected forms of a word so they can be analysed as a single item, identified by the word's lemma, or dictionary form. In computational linguistics, lemmati ... * Neurolemma, part of a neuron {{Disambiguation ...
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics i ...
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