Tverberg's theorem is a result in combinatorial geometry that concerns the division of points in Euclidean space. It states that for any set of \( (r-1)(d+1) + 1 \) points in \( \mathbb{R}^d \), it is possible to partition these points into \( r \) groups such that the \( r \) groups share a common point in their convex hulls.

The Wallace–Bolyai–Gerwien theorem is a result in geometry related to the transformation of polygons. Specifically, it states that any two simple polygons of equal area can be dissected into a finite number of polygonal pieces that can be rearranged to form one another. The theorem has important implications in the study of geometric dissections, a topic that has intrigued mathematicians for centuries.

In graph theory, a "lemma" is a proposition or statement that is proved and used as a stepping stone to prove a larger theorem. The term does not refer to a specific concept in graph theory itself but is rather a general mathematical term. Lemmas are commonly utilized to establish critical results or intermediate claims that help in constructing proofs of more significant theorems. They often simplify complex arguments by breaking them down into more manageable, verifiable pieces.

The BEST theorem, which stands for "Behavior of Extensibility under Sufficiently Tight Constraints," is a result in the field of fluid dynamics and elasticity, though it may also relate to other areas such as mathematical physics or control theory. However, the term "BEST theorem" is not widely recognized as a standard concept or theorem in mainstream mathematics or physics.

The Circle Packing Theorem is a result in mathematics that concerns arrangements of circles in a plane. Specifically, the theorem states that given any simple closed curve (a curve that does not intersect itself), it is possible to pack a finite number of circles within that curve such that all the circles are tangent to each other and to the curve.

The Erdős–Gallai theorem is a fundamental result in graph theory that pertains to the characterization of graphs with a given number of edges. Specifically, it provides a criterion for deciding whether a graph can exist with a specified number of edges and vertices, while also satisfying certain degree conditions.

The Erdős–Pósa theorem is a result in graph theory that deals with the relationship between the presence of certain subgraphs and the presence of certain structures in a graph. Specifically, it provides a relationship between the existence of a set of vertex-disjoint cycles in a graph and the existence of a set of vertices that intersects all these cycles. To state the theorem more formally, it addresses the case of cycles in graphs.

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).