Theorems in discrete mathematics

In discrete mathematics, a theorem refers to a statement that has been proven to be true based on previously established statements such as axioms, definitions, and other theorems. Theorems are integral to the field as they form the backbone of mathematical reasoning and structure. ### Key Components of Theorems in Discrete Mathematics: 1. **Definitions**: Before proving a theorem, precise definitions of terms involved are necessary to ensure clarity and avoid ambiguity.

In computational complexity theory, a theorem typically refers to a proven statement or result about the inherent difficulty of computational problems, particularly concerning the resources required (such as time or space) for their solution.

Blum's speedup theorem is a result in the field of computational complexity theory, specifically dealing with the relationship between the time complexity of algorithms and the computation of functions. Formulated by Manuel Blum in the 1960s, the theorem essentially asserts that if a certain function can be computed by a deterministic Turing machine within a certain time bound, then there exists an alternative algorithm (or Turing machine) that computes the same function more quickly.

The Cook–Levin theorem, established by Stephen Cook in 1971 and independently by Leonid Levin, is a fundamental result in computational complexity theory. It states that the Boolean satisfiability problem (SAT) is NP-complete. This means that SAT is at least as hard as any problem in the complexity class NP (nondeterministic polynomial time), and any problem in NP can be reduced to SAT in polynomial time.

Fagin's theorem is a fundamental result in the field of computational complexity theory, particularly concerning the classification of decision problems that can be expressed in terms of a certain type of logical formulas. Specifically, it characterizes the complexity of certain types of queries in databases. The theorem states that a decision problem is in the complexity class NP if and only if it can be expressed as a first-order logic formula with a quantifier prefix that allows for a fixed number of alternating quantifiers.

The Gap Theorem is a concept in the field of mathematics, particularly in the study of algebraic geometry and topology, though there are applications and related ideas in other areas of mathematics as well. In one of its forms, the Gap Theorem refers to a result concerning the existence of "gaps" in the spectrum of certain types of operators, particularly in the context of spectral theory.

The Karp–Lipton theorem is an important result in computational complexity theory that connects the complexity classes \(P\), \(NP\), and \(PSPACE\). It was established by Richard Karp and Richard J. Lipton in the early 1980s. The theorem states that if \(NP\) problems can be solved in polynomial time by a non-deterministic Turing machine using polynomial space (i.e.

The Linear Speedup Theorem is a concept in parallel computing that describes the efficiency of a parallel algorithm in relation to the number of processors used. Specifically, it states that if a problem can be perfectly parallelized, then using \( p \) processors can speed up the execution time by a factor of \( p \).

The Master Theorem is a powerful tool in the analysis of algorithms, particularly for solving recurrences that arise in divide-and-conquer algorithms. It provides a method for analyzing the time complexity of recursive algorithms without having to unroll the recurrence completely or use substitution methods.

The "No Free Lunch" (NFL) theorem in the context of search and optimization is a fundamental result that asserts that no optimization algorithm performs universally better than others when averaged over all possible problems. Introduced by David Wolpert and William Macready in the 1990s, the theorem highlights a crucial insight in the field of optimization and search algorithms. ### Key Concepts of the No Free Lunch Theorem 1.

The PCP (Probabilistically Checkable Proofs) theorem is a significant result in computational complexity theory that characterizes the class of decision problems that can be efficiently verified by a probabilistic verifier using a limited amount of randomness and reading only a small portion of the proof.

Savitch's theorem is a result in computational complexity theory that relates the complexity classes \( \text{NL} \) (nondeterministic logarithmic space) and \( \text{L} \) (deterministic logarithmic space).

Schaefer's Dichotomy Theorem is a result in the field of functional analysis, particularly in the study of nonlinear operators and fixed point theory. It provides a useful classification of certain types of operators in Banach spaces, particularly those that are continuous and compact.

The Sipser–Lautemann theorem is a result in the field of computational complexity theory that addresses the relationship between complexity classes, particularly focusing on the class of languages recognized by nondeterministic polynomial time machines (NP) and certain probabilistic polynomial time machines (BPP).

The Space Hierarchy Theorem is a fundamental result in computational complexity theory that pertains to the relationship between the space complexity of computational problems. It essentially states that there are problems that can be solved with a certain amount of space that cannot be solved with less space.

The Speedup Theorem is a concept from the field of computation and algorithms, particularly in the context of parallel computing and optimization. While there may be multiple interpretations or applications of the notion of speedup, one common formulation is related to how much faster an algorithm can run when resources are added (processing units, memory, etc.).

The Time Hierarchy Theorem is a fundamental result in computational complexity theory that formalizes the idea that more time allows for the solution of more problems. More specifically, it provides a rigorous framework for understanding how the class of problems that can be solved by deterministic Turing machines in polynomial time expands as the amount of time allowed increases.

Toda's theorem is a significant result in computational complexity theory, which establishes a relationship between different complexity classes.

The Valiant–Vazirani theorem is a result in theoretical computer science concerning the complexity of certain problems in the context of randomized algorithms. Specifically, it relates to the complexity class NP (nondeterministic polynomial time) and the concept of zero-knowledge proofs.

In graph theory, a branch of mathematics that deals with the study of graphs, which are structures used to model pairwise relations between objects, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, like axioms. There are many important theorems in graph theory, each contributing to our understanding of graphs and their properties.

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 2-factor theorem, also known as the two-factor theorem, is a principle in number theory and algebra related to the factorization of polynomials or functions. Specifically, it states that if a polynomial can be factored into two non-trivial factors, then the degree of one factor and the degree of the other factor must sum to the degree of the original polynomial.

Alspach's conjecture, proposed by Alspach in 1970, is a conjecture in the field of graph theory. It pertains to the existence of certain types of graphs known as 1-factorizations of complete graphs.

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.

Cederbaum's maximum flow theorem is a specific result in the field of network flow theory. It provides a condition under which the maximum flow in a flow network equals the minimum cut capacity. This theorem can be particularly useful in understanding the limits of flow through networks and in applications across various fields like computer science, operations research, and telecommunications.

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 Even Circuit Theorem, often referred to in the context of graph theory and circuit design, primarily deals with the properties of circuits within graphs. While the term itself may not be universally defined across all disciplines, it is likely related to concepts in electrical engineering and theoretical computer science, where circuits can be represented as graphs. In general terms, in a graph: - A circuit (or cycle) is a closed path where no edges are repeated.

Fleischner's theorem is a result in graph theory that relates to the properties of cycles in Eulerian graphs. Specifically, it states that every 2-edge-connected graph (a graph where there are at least two vertex-disjoint paths between any two vertices) contains a cycle that includes every edge of the graph. This is closely associated with the concept of an Eulerian circuit, which is a cycle that visits every edge of a graph exactly once.

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 Gale–Ryser theorem is a result in combinatorial mathematics, specifically in the theory of bipartite graphs and matching. It provides a characterization of the matchings in bipartite graphs based on certain conditions related to degree sequences.

The Geiringer–Laman theorem is a result in the field of graph theory and combinatorial geometry, specifically concerning the rigidity of frameworks. The theorem provides a criterion for determining when a certain kind of graph, known as a "framework", can be considered rigid, meaning that its vertices cannot be moved without distorting the distances between them.

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.

Kotzig's theorem is a result in graph theory concerning the properties of certain types of graphs, particularly related to edge colorings. Specifically, it states that every connected graph with a minimum degree of at least 3 can be decomposed into two spanning trees. This result is significant because spanning trees are foundational structures in graph theory, and their decomposition has implications for network design and reliability.

Kuratowski's theorem is a fundamental result in topology, specifically in the area of planar graphs. It characterizes the planarity of a graph using the concepts of subgraphs and Kuratowski's two examples of non-planar graphs.

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.

Ore's theorem is a result in graph theory concerning the conditions under which a graph is Hamiltonian, meaning that it contains a Hamiltonian circuit (a cycle that visits every vertex exactly once).

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 Planar Separator Theorem is a concept in computational geometry and graph theory which states that for any planar graph, it is possible to partition the vertices of the graph into three disjoint sets: X, Y, and S. The sets have the following properties: 1. **Small Separator Size**: The size of the set S (the separator) is proportional to the square root of the number of vertices in the graph.

Robbins' theorem is a significant result in the field of Boolean algebra and combinatorial logic, primarily related to the minimization of Boolean functions. The theorem, formulated by Howard Robbins in 1937, states that any boolean function can be represented using a certain set of logical operations. Specifically, it provides a characterization of boolean functions that can be expressed using certain combinations of the logical operations AND, OR, and NOT.

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.

The Strong Perfect Graph Theorem, proved by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas in 2006, establishes an important characterization of perfect graphs. The theorem states that a graph is perfect if and only if it contains no induced subgraph that is an odd cycle of length at least 5 or the complement of such a cycle (i.e., a complete graph minus an odd cycle).

Turán's theorem is a fundamental result in extremal graph theory that provides a bound on the number of edges in a graph that avoids complete subgraphs (cliques) of a given size. Specifically, it deals with the maximum number of edges that can be present in a graph with \( n \) vertices that does not contain a complete subgraph \( K_{r+1} \) (a complete graph on \( r+1 \) vertices).

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.

Wagner's theorem is a result in graph theory that provides a characterization of planar graphs. Specifically, it states that a graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph \( K_{5} \) (the complete graph on five vertices) or a subdivision of the complete bipartite graph \( K_{3,3} \) (the complete bipartite graph with three vertices in each part).

The Akra–Bazzi method is a technique used in the analysis of the time complexity of divide-and-conquer algorithms. It provides a systematic way to solve recurrence relations of the form: \[ T(n) = g(n) + \sum_{i=1}^{k} T\left( \frac{n}{b_i} \right) \] where: - \( T(n) \) is the time complexity we want to solve.

The Analyst's Traveling Salesman Theorem is a result in the field of real analysis, specifically in the context of metric spaces and geometry of numbers. It addresses the existence of paths that can be constructed in a certain way, related to the traveling salesman problem.

The Bregman–Minc inequality relates to matrix theory and provides a bound on the determinants of matrices. It is a useful result in the context of matrix analysis, particularly concerning positive semidefinite matrices.

Friedman's SSCG (Stochastic Simulation and Control Game) function is a concept used in the context of economics and decision theory, particularly related to dynamic programming and optimal control. The SSCG function is often utilized to model and analyze strategic interactions and decisions under uncertainty. The exact formulation of the SSCG function can vary, but it typically involves aspects of stochastic processes, where outcomes depend not only on the current state and action but also on random events that can influence future states.

Holland's Schema Theorem is a foundational concept in the field of genetic algorithms, introduced by John Holland in his book "Adaptation in Natural and Artificial Systems" published in 1975. The theorem provides a theoretical framework for understanding how genetic algorithms evolve solutions over time. ### Key Concepts of Holland's Schema Theorem: 1. **Schema**: A schema is a template that represents a subset of strings with similarities at certain positions and wildcards (denoted by `*`) at others.

Kruskal's tree theorem is a result in graph theory and combinatorics that deals with the structure of trees and their embeddings within each other. More specifically, it provides criteria for the comparison and embedding of trees.