For a more detailed and technical article, see Ramsey's theorem.

See the article on Ramsey's theorem for a rigorous proof.

This also is a special case of Ramsey's theorem, which says that for any given integer c, any given integers n < sub > 1 </ sub >,..., n < sub > c </ sub >, there is a number, R ( n < sub > 1 </ sub >,..., n < sub > c </ sub >), such that if the edges of a complete graph of order R ( n < sub > 1 </ sub >,..., n < sub > c </ sub >) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order n < sub > i </ sub > whose edges are all colour i. The special case above has c

Many theorems, which are modeled after Ramsey's theorem itself, assert that in every partition of a large structured object, one of the classes necessarily contains a large structured subobject, but give no information about which class this is.

In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs.

For two colours, Ramsey's theorem states that for any pair of positive integers ( r, s ), there exists a least positive integer R ( r, s ) such that for any complete graph on R ( r, s ) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red.

Ramsey's theorem is a foundational result in combinatorics.

The numbers R ( r, s ) in Ramsey's theorem ( and their extensions to more than two colours ) are known as Ramsey numbers.

# REDIRECT Ramsey's theorem

# REDIRECT Ramsey's theorem

* Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number of vertices.

# redirect Ramsey's theorem # Ramsey numbers

In 1930, in a paper entitled ' On a Problem in Formal Logic ,' Frank P. Ramsey proved a very general theorem ( now known as Ramsey's theorem ) of which this theorem is a simple case.

Metro declined to appeal Judge Ramsey's latest ruling, and the Prince George's County Council voted to reverse its earlier decision and support the original Green Line route to Branch Avenue.

* Biographical information, gubernatorial records, and Ramsey's personal papers are available for research use at the Minnesota Historical Society.

* Archbishop Ramsey's papers are held at Lambeth Palace Library

He is killed by one of the infected, once it discovers the Exiles are trying to create a cure based on Ramsey's original strain of the virus.

They failed in qualification for the 1974, leading to Alf Ramsey's dismissal, and 1978 World Cups.

The Germans had hope now, especially when Beckenbauer became more liberated in the game with Ramsey's decision to substitute Bobby Charlton.

Hurst settled into international football quickly but as the World Cup approached, it seemed clear that his inclusion in Ramsey's squad of 22 would merely be as a different option to the first choice partnership of Jimmy Greaves and Roger Hunt.

Yet another version of deflationism is the prosentential theory of truth, first developed by Dorothy Grover, Joseph Camp, and Nuel Belnap as an elaboration of Ramsey's claims.

Ramsey's area of expertise is intelligence and counter-intelligence.

Ramsey's somewhat slurred speech, a trademark of her later performances, was caused in part from having had some of her tongue and her jaw removed during surgery for esophageal cancer in 1984.

In 1988, Ramsey's cancer returned.

NJ Route 17 and County Route 507 intersect the areas east and north of Ramsey's downtown business district, while I-287 and U. S. Route 202 pass through the Darlington section of Mahwah to the west and the New York State Thruway ( I-87 / I-287 ) and NY Route 59 run through Suffern, New York to the north.

In the 1960s, the capitol was reconstructed, based largely on the dimensions given in historian J. G. M. Ramsey's Annals of Tennessee.

In Byrhtferth of Ramsey's Life of Saint Oswald, Oda is said to have joined the household of a pious nobleman called Æthelhelm, whom he accompanied to Rome on pilgrimage.

The teenager was soon hailed as a possible late call-up for Alf Ramsey's 1966 World Cup squad, having been included in the original 40-man squad announced in April 1966, but he was not included in the final 22.

Line of sight distance d of this station is given by the Pythagorean theorem ;

* The Murderous Maths of Everything, ISBN 1-407-10367-9 ( prime numbers, Sieve of Eratosthenes, Pythagoras ' Theorem, triangle numbers, square numbers, the International Date Line, geometry, geometric constructions, topology, Mobius strips, curves ( conic sections and cycloids Golomb Rulers, 4 dimensional " Tic Tac Toe ", The Golden Ratio, Fibonacci sequence, Logarithmic spirals, musical ratios, Theorems ( including Ham sandwich theorem and Fixed point theorem ), probability ( cards, dice etc.

** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be graph coloring | colored with four color theorem | only four colors.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

An easier to state version of the theorem uses graph theory.

In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, " every planar graph is four-colorable " (; ).

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

* The strong perfect graph theorem

The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X ; but every such Y is again a graph.

Cayley's formula is the special case of complete graphs in a more general problem of counting spanning trees in an undirected graph, which is addressed by the matrix tree theorem.

The mean value theorem proves that this must be true: The slope between any two points on the graph of f must equal the slope of one of the tangent lines of f. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero.

Dunwoody found a graph-theoretic proof of Stallings ' theorem about ends of groups in 1982, by constructing certain tree-like automorphism invariant graph decompositions.

The resulting Tutte graph is 3-connected and planar, so by Steinitz ' theorem it is the graph of a polyhedron.

* Grinberg's theorem, a necessary condition on the existence of a Hamiltonian cycle that can be used to show that a graph forms a counterexample to Tait's conjecture

BCT1 is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.