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In combinatorics, Ramsey's theorem states that in colouring a large complete graph (that is a simple graph, where an edge connects every pair of vertices), one will find complete subgraphs all of the same colour. For 2 colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists an 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. Here R(r,s) signifies an integer that depends on both r and s. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...
In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ...
This article just presents the basic definitions. ...
This article just presents the basic definitions. ...
This article just presents the basic definitions. ...
In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ...
Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory, now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of monochromatic subsets, that is, subsets of connected edges of just one colour. Frank Plumpton Ramsey (February 22, 1903 - January 19, 1930) was a British mathematician and logician. ...
Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ...
An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours c, and any given integers n1,...,nc, there is a number, R(n1, ..., nc), such that if the edges of a complete graph of order R(n1, ..., nc) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order ni whose edges are all colour i. The special case above has c = 2 (and n1 = r and n2 = s).
Example: R(3,3)=6
 A 2-coloring of K5 with no monochromatic K3 Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex v. There are 5 edges incident to v and so (by the pigeonhole principle) at least 3 of them must be the same colour. Without loss of generality we can assume at least 3 of these edges, connecting to vertices r, s and t, are blue. (If not, exchange red and blue in what follows.) If any of the edges (r, s), (r, t), (s, t) are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have with an entirely red triangle. Since this argument works for any colouring, any K6 contains a monochromatic K3, and therefore that R(3,3) ≤ 6. The popular version of this is called the theorem on friends and strangers. Diagram of a 2-coloured K_5 without a monochromatic K_3. ...
The inspiration for the name of the principle: pigeons in holes. ...
Without loss of generality or simply WLOG is a frequently used expression in mathematics. ...
The friendship theorem is a mathematical theorem in an area of mathematics called Ramsey theory. ...
An alternate proof works by double counting. It goes as follows: Count the number of ordered triples of vertices x, y, z such that the edge (xy) is red and the edge (yz) is blue. Firstly, any given vertex will be the middle of either 0×5=0, 1×4=4 or 2 × 3 = 6 such triples. Therefore there are at most 6×6=36 such triples. Secondly, for any non-monochromatic triangle (xyz), there exist precisely two such triples. Therefore there are at most 18 non-monochromatic triangles. Therefore there are at least 2 monochromatic triangles. In combinatorics, double counting, also called two-way counting, is a proof technique that involves counting the size of a set in two ways in order to show that the two resulting expressions for the size of the set are equal. ...
Conversely, it is possible to 2-colour a K5 without creating any monochromatic K3, showing that R(3,3) > 5. The unique colouring is shown to the right. Thus R(3,3) = 6.
Proof of the theorem We prove the theorem for the 2-colour case, by induction on r + s. It is clear from the definition that for all n, R(n, 1) = R(1, n) = 1. This starts the induction. We prove that R(r, s) exists by finding an explicit bound for it. By the inductive hypothesis R(r − 1, s) and R(r, s − 1) exist. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ...
Claim: R(r, s) ≤ R(r − 1, s) + R(r, s − 1): Consider a complete graph on R(r − 1, s) + R(r, s − 1) vertices. Pick a vertex v from the graph and consider two subgraphs M and N where a vertex w is in M if and only if (v, w) is blue and is in N otherwise. It has been suggested that this article or section be merged with Logical biconditional. ...
Now |M| ≥ R(r − 1, s) or |N| ≥ R(r, s − 1), again by the pigeonhole principle. In the former case if M has a red Ks then so does the original graph and we are finished. Otherwise M has a blue Kr−1 and so M union {v} has blue Kr by definition of M. The latter case is analogous.
Thus the claim is true and we have completed the proof for 2 colours. We now prove the result for the general case of c colours. The proof is again by induction, this time on the number of colours c. We have the result for c = 1 (trivially) and for c = 2 (above). Now let c > 2.
Claim: R(n1, ..., nc) ≤ R(n1, ..., nc−2, R(nc− 1, nc))
Note, that the right hand side only contains Ramsey numbers for c− 1 or 2 colours, and therefore exists and is finite number t, by the inductive hypothesis. Thus, proving the claim will prove the theorem.
Proof of claim: Consider a graph on t vertices and colour its edges with c colours. Now 'go colour-blind' and pretend that c − 1 and c are the same colour. Thus the graph is now (c − 1)-coloured. By the inductive hypothesis, it contains either a Kni monochromatically coloured with colour i for some 1 ≤ i ≤ (c − 2) or a KR(nc−1,nc)-coloured in the 'blurred colour'. In the former case we are finished. In the latter case, we recover our sight again and see from the definition of R(nc−1, nc) we must have either a (c − 1)-monochrome Knc−1 or a c-monochrome Knc. In either case the proof is complete.
Ramsey numbers The numbers R(r,s) in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. An upper bound for R(r,s) can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first lower bound was proved by Paul Erdos using the probabilistic method.)However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. Consequently, there are very few numbers r and s for which we know the exact value of R(r,s). Computing a lower bound L for R(r,s) usually requires exhibiting a blue/red colouring of the graph KL−1 with no blue Kr subgraph and no red Ks subgraph. Searching all colourings of a graph Kn becomes computationally extremely difficult as n increases; the number of colourings grows super-exponentially. Paul Erdos Paul ErdÅ‘s (March 26, 1913 – September 20, 1996) was an immensely prolific and famously eccentric mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory. ...
This article is not about probabilistic algorithms, which give the right answer with high probability but not with certainty, nor about Monte Carlo methods, which are simulations relying on pseudo-randomness. ...
In mathematics, a quantity that grows exponentially is one whose growth rate is always proportional to its current size. ...
At the time of writing, even the exact value of R(5,5) is unknown, although it is known to lie between 43 and 49 (inclusive); barring a breakthrough in theory, it is probable that the exact value of R(6,6) will remain unknown forever.
- "Imagine an alien force, vastly more powerful than us landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they asked for R(6, 6), we should attempt to destroy the aliens". - Paul Erdős
R(r,s) for values of r and s up to 10 are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. R(r,s) for values of r and s less than 3 are given by R(1,s) = 1 and R(2,s) = s for all values of s. The standard survey on the development of Ramsey number research has been written by Stanisław Radziszowski, who also found the exact value of R(4,5) (with Brendan McKay). Paul ErdÅ‘s also Pál ErdÅ‘s, in English Paul Erdos or Paul Erdös, (March 26, 1913 – September 20, 1996) was an immensely prolific (and famously eccentric) Hungarian mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set...
Stanisław Radziszowski - Currently a professor of Computer Science at the Rochester Institute of Technology since 1984, Prof. ...
Brendan D. McKay is a Professor in the Department of Computer Science at ANU (Australian National University). ...
| r,s | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | | 2 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | 3 | 1 | 3 | 6 | 9 | 14 | 18 | 23 | 28 | 36 | 40–43 | | 4 | 1 | 4 | 9 | 18 | 25 | 35–41 | 49–61 | 56–84 | 69–115 | 92–149 | | 5 | 1 | 5 | 14 | 25 | 43–49 | 58–87 | 80–143 | 101–216 | 121–316 | 141–442 | | 6 | 1 | 6 | 18 | 35–41 | 58–87 | 102–165 | 111–298 | 127–495 | 169–780 | 178–1171 | | 7 | 1 | 7 | 23 | 49–61 | 80–143 | 111–298 | 205–540 | 216–1031 | 232–1713 | ≤ 2826 | | 8 | 1 | 8 | 28 | 56–84 | 101–216 | 127–495 | 216–1031 | 282–1870 | 317–3583 | ≤ 6090 | | 9 | 1 | 9 | 36 | 69–115 | 121–316 | 169–780 | 232–1713 | 317–3583 | 565–6588 | 580–12677 | | 10 | 1 | 10 | 40–43 | 92–149 | 141–442 | 178–1171 | ≤ 2826 | ≤ 6090 | 580–12677 | 798–23556 |
A Multicolor Example: R(3,3,3) = 17
 The only two 3-coloring of K 16 with no monochromatic K 3. The untwisted coloring (top) and the twisted coloring (bottom). A multicolor Ramsey number is a Ramsey number using 3 or more colors. There is only one multicolor Ramsey number for which the exact value is known, namely R(3,3,3) = 17. Image File history File links Download high-resolution version (612x792, 537 KB) I originally created this image using MetaPost and latex, resulting in a pdf. ...
Image File history File links Download high-resolution version (612x792, 537 KB) I originally created this image using MetaPost and latex, resulting in a pdf. ...
Suppose that you have an edge coloring of a complete graph using 3 colors, red, yellow and green. Suppose further that the edge coloring has no monochromatic triangles. Select a vertex v. Consider the set of vertices which have a green edge to the vertex v. This is called the green neighborhood of v. The green neighborhood of v cannot contain any green edges, since otherwise there would be a green triangle consisting of the two endpoints of that green edge and the vertex v. Thus, the induced edge coloring on the green neighborhood of v has edges colored with only two colors, namely yellow and red. Since R(3,3) = 6, the green neighborhood of v can contain at most 5 vertices. Similarly, the red and yellow neighborhoods of v can contain at most 5 vertices each. Since every vertex, except for v itself, is in one of the green, red or yellow neighborhoods of v, the entire complete graph can have at most 1 + 5 + 5 + 5 = 16 vertices. Thus, we have R(3,3,3) ≤ 17.
To see that R(3,3,3) ≥ 17, it suffices to draw an edge coloring on the complete graph on 16 vertices with 3 colors, which avoids monochromatic triangles. It turns out that there are exactly two such colorings on K16, the so-called untwisted and twisted colorings. Both colorings are shown in the figure to the right, with the untwisted coloring on the top, and the twisted coloring on the bottom. In both colorings in the figure, note that the vertices are labeled, and that the vertices v11 through v15 are drawn twice, on both the left and the right, in order to simplify the drawings.
Thus, R(3,3,3) = 17.
If you select any color of either the untwisted or twisted coloring on K16, and consider the graph whose edges are precisely those edges which have the specified color, you will get the Clebsch Graph. Image File history File links Download high-resolution version (725x696, 75 KB) Clebsch graph. ...
Image File history File links Download high-resolution version (725x696, 75 KB) Clebsch graph. ...
It is known that there are exactly two edge colorings with 3 colors on K15 which avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colorings on K16, respectively.
It is also known that there are exactly 115 edge colorings with 3 colors on K14 which avoid monochromatic triangles, provided that we consider edge colorings which differ by a permutation of the colors as being the same.
Extensions of the theorem The theorem can also be extended to hypergraphs. An m-hypergraph is a graph whose "edges" are sets of m vertices - in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers m and c, and any integers n1,...,nc, there is an integer R(n1,...,nc;c,m) such that if the hyperedges of a complete m-hypergraph of order R(n1,...,nc;c,m) are coloured with c different colours, then for some i between 1 and c, the hypergraph must contain a complete sub-m-hypergraph of order ni whose hyperedges are all colour i. This theorem is usually proved by induction on m, the 'hyper-ness' of the graph. The base case for the proof is m=2, which is exactly the theorem above. In mathematics, a hypergraph is a generalization of a graph, where edges can connect any number of vertices. ...
Infinite Ramsey theory A further result, also commonly called Ramsey's theorem, applies to infinite graphs. In a context where finite graphs are also being discussed it is often called the "Infinite Ramsey theorem". As intuition provided by the pictorial representation of a graph is diminished when moving from finite to infinite graphs, theorems in this area are usually phrased in set-theoretic terminology. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
Theorem: Let X be some countably infinite set and colour the elements of X(n) (the subsets of X of size n) in c different colours. Then there exists some infinite subset M of X such that the size n subsets of M all have the same colour. In mathematics, a countable set is a set with the same cardinality (i. ...
Proof: The proof is given for c=2. It is easy to prove the theorem for an arbitrary number of colours using a 'colour-blindness' argument as above. The proof is by induction on 'n', the size of the subsets. For n = 1,the statement is equivalent to saying that if you split an infinite set into two sets, one of them is infinite. This is evident. Assuming the theorem is true for n ≤ r, we prove it for n = r + 1. Given a 2-colouring of the (r + 1)-element subsets of infinite set X, choose element a0 of X (note that in the statement of the theorem we insisted that X was countably infinite, if X were a general infinite set we would require the axiom of choice to make this choice) and let Y = Xa0. We then induce a 2-colouring of the r-element subsets of Y, by just adding a0 to each r-element subset (to get an (r+1)-element subset of X). By the induction hypothesis, there exists an infinite subset Y1 within Y such that every r-element subset of Y is coloured the same colour in the induced colouring. Therefore we have chosen an element a0 and a subset Y1 such that every (r+1)-element subset of X consisting of a0 and r elements of Y1 has the same colour. Continuing in this we can choose a1 from Y1 and subset Y2 of Y1 with the same properties. We end with a sequence {a0,a1,a2,...} such that the colour of each (r + 1)-element subset (ai(1),ai(2),...,ai(r+1)) with i(1) < i(2) < ... < i(r + 1) depends only on the value of i(1). Further, there are infinitely many values of i(n) such that this colour will be the same. Take these ai(n)'s to get the desired monochromatic set. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
Infinite version implies the finite It is easy to deduce the finite Ramsey theorem from the infinite one using a proof by contradiction. Suppose the finite Ramsey Theorem is false. Then there exists c,n,T such that for every integer k, there exists a c-colouring of [k](n), without a monochromatic set of size T. Let Ck denote the c-colourings of [k](n) without a monochromatic set of size T. Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument...
For any integer k, given any colouring in Ck + 1, if we restrict the colouring to [k](n) (by ignoring the colour of all sets containing k + 1), then we get a colouring in Ck. Define  to be the colourings in Ck which are restrictions of colourings in Ck + 1. Since Ck + 1 is not empty, nor is .
Similarly, the restriction of any colouring in  is in , allowing us to define  as the set of all such restrictions, which we can see is not empty. Continue doing so, defining  for all integers n,k.
Now, for any integer k,  , and each set is non-empty. Furthermore,  , and hence Ck is finite. It follows that the intersection of all of these sets must be non-empty. Let  . Then everything in Dk is the restriction of something in Dk + 1. Therefore we can start with something in Dn, and unrestrict to something in Dn + 1, and continue doing so, to get a colouring of  without a monochromatic set of size T.
If we take a suitable topological viewpoint, this argument becomes a standard compactness argument showing that the infinite version of the theorem implies the finite version. The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i. ...
Directed graph Ramsey numbers It is also possible to define Ramsey numbers for directed graphs. (These were introduced by P.Erdos & L.Moser.) Let R(n) be the smallest number Q such that any complete graph with singly-directed arcs (also called a "tournament") and with ≥Q nodes contains an acyclic (also called "transitive") n-node subtournament.
This is the directed-graph analogue of what (above) has been called R(n,n;2), the smallest number Z such that any 2-colouring of the edges of a complete undirected graph with ≥Z nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc colors is the two directions of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")
Indeed many find the directed graph problem to actually be more elegant than the unidirected one. We have R(0)=0, R(1)=1, R(2)=2, R(3)=4, R(4)=8, R(5)=14, R(6)=28, 32≤R(7)≤55, and R(8) is again a problem you do not want powerful aliens to pose.
References - F. P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. series 2, vol. 30 (1930), pp. 264-286
See also In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory is true but not provable in Peano arithmetic. ...
This article is about a pencil game. ...
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