The Erdős–Stone theorem is a fundamental result in extremal graph theory, which deals with understanding the maximum number of edges in a graph that does not contain a particular subgraph. Specifically, the theorem provides a way to determine the asymptotic behavior of the maximum number of edges in a graph on \( n \) vertices that does not contain a complete subgraph \( K_r \) (the complete graph on \( r \) vertices) as a subgraph.

The Fulkerson–Chen–Anstee theorem is a result in graph theory, particularly related to the field of perfect graphs. The theorem establishes that certain properties hold for certain types of graphs, specifically focusing on the behavior of graph complements and their chromatic numbers. The theorem is often framed in the context of *perfect graphs*, which are defined as graphs where the chromatic number of the graph equals the size of the largest clique in the graph for every induced subgraph.

Fáry's theorem is a result in the field of graph theory that states that every simple planar graph can be embedded in the plane such that its edges are represented as straight-line segments. In simpler terms, it asserts that for any graph that can be drawn on a plane without any edges crossing (i.e., it is planar), there exists a way to draw it in the same plane where all edges are straight lines.

The Graph Structure Theorem is a significant result in graph theory that characterizes certain classes of graphs. Specifically, it provides a structural decomposition of a broad class of graphs known as "H-minor-free graphs." This theorem states that if a graph does not contain a fixed graph H as a minor, then it can be decomposed into a bounded number of simpler components that exhibit certain structural properties.

Grinberg's theorem is a result in the field of topology and specifically pertains to the properties of continuous mappings between topological spaces. It is often mentioned in the context of compact spaces and homeomorphisms. The theorem states that if \( X \) is a compact Hausdorff space and \( Y \) is a connected space, then every continuous surjective mapping from \( X \) onto \( Y \) is a quotient map.

The Max-flow Min-cut Theorem is a fundamental result in network flow theory, specifically in the context of directed (or undirected) graphs. It provides a deep relationship between two concepts: the maximum amount of flow that can be sent from a source node to a sink node in a flow network and the minimum capacity that, when removed, would disconnect the source from the sink.

The Perfect Graph Theorem is a result in graph theory that characterizes perfect graphs. A graph is considered *perfect* if, for every induced subgraph, the chromatic number (the smallest number of colors needed to color the graph such that no two adjacent vertices share the same color) equals the size of the largest clique (a subset of vertices, all of which are adjacent to each other).

The number 2 is a natural number that follows 1 and precedes 3. It is an even integer and is the smallest and the first prime number. In various contexts, 2 can represent a quantity, a position, or a score, among other things.

The Robertson–Seymour theorem, a significant result in graph theory, is a foundational result in the study of graph minors. Formulated by Neil Robertson and Paul D. Seymour in a groundbreaking series of papers from the late 20th century, the theorem states that: **Any minor-closed family of graphs can be characterized by a finite set of forbidden minors.

Schnyder's theorem, or Schnyder's realizability theorem, is a result in graph theory that relates to the representation of planar graphs. It states that: **Every simple planar graph can be embedded in the plane such that its vertices can be labeled with numbers from {0, 1, 2, 3} so that the edges of the graph respect certain ordering conditions.

Veblen's theorem is a result in the field of set theory and topology, specifically in the context of the study of properties of certain sets. It primarily deals with the concept of "well-ordering." The theorem states that every set can be well-ordered, meaning that its elements can be arranged in a sequence such that every non-empty subset has a least element.

Armin Moczek is an American evolutionary biologist known for his research on the evolution of morphological diversity, particularly in the context of insect development and adaptive radiation. He is a professor at Indiana University and has contributed significantly to the field through studies on the evolution of traits in organisms, including the role of genetic and ecological factors in shaping diversity. Moczek's work often involves the use of model organisms, such as beetles, to explore the underlying mechanisms of evolutionary change.

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided that the system has a unique solution. It is applicable when the coefficient matrix is non-singular (i.e., its determinant is non-zero).

The Edge-of-the-Wedge theorem is a concept from complex analysis, specifically regarding holomorphic functions. It deals with the behavior of these functions on regions in the complex plane that have "wedge-shaped" domains.

The Mermin-Wagner theorem is a result in statistical mechanics and condensed matter physics that addresses the behavior of certain types of physical systems at low temperatures, specifically those defined by continuous symmetry. The theorem, which was formulated by N. D. Mermin and H. Wagner in the 1960s, states that in two-dimensional systems with continuous symmetry, spontaneous symmetry breaking and long-range order cannot occur at finite temperatures.

Theorems about polygons constitute a significant part of geometry, focusing on the properties, relationships, and characteristics of various types of polygons.

MacMahon's Master Theorem is a mathematical tool used in the analysis of combinatorial structures, particularly in the enumeration of various combinatorial objects. While it's not as widely known as some other results in combinatorics, it provides a framework for counting partitions, arrangements, and related structures using generating functions. The theorem is named after the British mathematician Percy MacMahon, who made significant contributions to the theory of partitions and generating functions.

The Principal Axis Theorem, often discussed in the context of linear algebra and quadratic forms, refers to a method of diagonalizing a symmetric matrix. This theorem states that for any real symmetric matrix, there exists an orthogonal matrix \(Q\) such that: \[ Q^T A Q = D \] where \(A\) is the symmetric matrix, \(Q\) is an orthogonal matrix (i.e.