Ramsey's theorem


In combinatorial mathematics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling of a sufficiently large complete graph. To demonstrate the theorem for two colours, let r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on r vertices or a red clique on s 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.
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, such that if the edges of a complete graph of order R 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.

Example: ''R''(3, 3) = 6

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 at least 3 of them must be the same colour. Without loss of generality we can assume at least 3 of these edges, connecting the vertex, v, to vertices, r, s and t, are blue. If any of the edges,,,, are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Since this argument works for any colouring, any K6 contains a monochromatic K3, and therefore R ≤ 6. The popular version of this is called the theorem on friends and strangers.
An alternative proof works by double counting. It goes as follows: Count the number of ordered triples of vertices, x, y, z, such that the edge,, is red and the edge,, 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, there exist precisely two such triples. Therefore, there are at most 18 non-monochromatic triangles. Therefore, at least 2 of the 20 triangles in the K6 are monochromatic.
Conversely, it is possible to 2-colour a K5 without creating any monochromatic K3, showing that R > 5. The unique colouring is shown to the right. Thus R = 6.
The task of proving that R ≤ 6 was one of the problems of William Lowell Putnam Mathematical Competition in 1953, as well as in the Hungarian Math Olympiad in 1947.

Proof of the theorem

2-colour case

The theorem for the 2-colour case, can be proved by induction on. It is clear from the definition that for all, =. This starts the induction. We prove that exists by finding an explicit bound for it. By the inductive hypothesis and exist.
Proof. Consider a complete graph on vertices whose edges are coloured with two colours. Pick a vertex from the graph, and partition the remaining vertices into two sets and, such that for every vertex, is in if is blue, and is in if is red. Because the graph has = vertices, it follows that either or. In the former case, if has a red then so does the original graph and we are finished. Otherwise has a blue and so has a blue by the definition of. The latter case is analogous. Thus the claim is true and we have completed the proof for 2 colours.
In this 2-colour case, if and are both even, the induction inequality can be strengthened to:
Proof. Suppose and are both even. Let and consider a two-coloured graph of vertices. If is degree of -th vertex in the graph, then, according to the Handshaking lemma, is even. Given that is odd, there must be an even. Assume is even, and are the vertices incident to vertex 1 in the blue and red subgraphs, respectively. Then both and are even. According to the Pigeonhole principle, either, or. Since is even, while is odd, the first inequality can be strengthened, so either or. Suppose. Then either the subgraph has a red and the proof is complete, or it has a blue which along with vertex 1 makes a blue. The case is treated similarly.

Case of more colours

Lemma 2. If c>2, then R ≤ R.
Proof. Consider a complete graph of R 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 -coloured. Due to the definition of R, such a graph contains either a Kni mono-chromatically coloured with colour i for some 1 ≤ i ≤ c − 2 or a KR-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 we must have either a -monochrome Knc−1 or a c-monochrome Knc. In either case the proof is complete.
Lemma 1 implies that any R is finite. The right hand side of the inequality in Lemma 2 expresses a Ramsey number for c colours in terms of Ramsey numbers for fewer colours. Therefore any R is finite for any number of colours. This proves the theorem.

Ramsey numbers

The numbers in Ramsey's theorem are known as Ramsey numbers. The Ramsey number,, gives the solution to the party problem, which asks the minimum number of guests,, that must be invited so that at least will know each other or at least will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices,, such that all undirected simple graphs of order, contain a clique of order, or an independent set of order. Ramsey's theorem states that such a number exists for all and.
By symmetry, it is true that. An upper bound for can be extracted from the proof of the theorem, and other arguments give lower bounds. However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. There are also very few numbers and for which we know the exact value of.
Computing a lower bound for usually requires exhibiting a blue/red colouring of the graph with no blue subgraph and no red subgraph. Such a counterexample is called a Ramsey graph. Brendan McKay maintains a list of known Ramsey graphs. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence. A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. Each complete graph has edges, so there would be a total of graphs to search through if brute force is used. Therefore, the complexity for searching all possible graphs is for colourings and an upper bound of nodes.
As described above,. It is easy to prove that, and, more generally, that for all : a graph on nodes with all edges coloured red serves as a counterexample and proves that ; among colourings of a graph on nodes, the colouring with all edges coloured red contains a -node red subgraph, and all other colourings contain a -node blue subgraph
Using induction inequalities, it can be concluded that, and therefore. There are only two graphs among different -colourings of -node graphs, and only one graph among colourings. It follows that.
The fact that was first established by Brendan McKay and Stanisław Radziszowski in 1995.
The exact value of is unknown, although it is known to lie between and .
In 1997, McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture that. They were able to construct exactly 656 graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a graph.
For with, only weak bounds are available. Lower bounds for and have not been improved since 1965 and 1972, respectively.
with are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. with are given by and for all values of. The standard survey on the development of Ramsey number research is the Dynamic Survey 1 of the Electronic Journal of Combinatorics. It was written and is updated by Stanisław Radziszowski. Its latest update was in March 2017. In general, there are a few years between the updates.
Where not cited otherwise, entries in the table below are taken from this dynamic survey. Note there is a trivial symmetry across the diagonal since.
12345678910
1
2
3
4
5
6
7
8
9
10

Asymptotics

The inequality may be applied inductively to prove that
In particular, this result, due to Erdős and Szekeres, implies that when,
An exponential lower bound,
was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is obviously a huge gap between these two bounds: for example, for, this gives. Nevertheless, exponential growth factors of either bound have not been improved to date and still stand at and respectively. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers currently stand at
due to Spencer and Conlon respectively.
For the off-diagonal Ramsey numbers, it is known that they are of order ; this may be stated equivalently as saying that the smallest possible independence number in an -vertex triangle-free graph is
The upper bound for is given by Ajtai, Komlós, and Szemerédi, the lower bound was obtained originally by Kim, and was improved by Griffiths, Morris, Fiz Pontiveros, and Bohman and Keevash, by analysing the triangle-free process. More generally, for off-diagonal Ramsey numbers,, with fixed and growing, the best known bounds are
due to Bohman and Keevash and Ajtai, Komlós and Szemerédi respectively.

A multicolour example: ''R''(3, 3, 3) = 17

A multicolour Ramsey number is a Ramsey number using 3 or more colours. There are only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely R = 17 and R = 30.
Suppose that we have an edge colouring of a complete graph using 3 colours, red, green and blue. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex v. Consider the set of vertices that have a red edge to the vertex v. This is called the red neighbourhood of v. The red neighbourhood of v cannot contain any red edges, since otherwise there would be a red triangle consisting of the two endpoints of that red edge and the vertex v. Thus, the induced edge colouring on the red neighbourhood of v has edges coloured with only two colours, namely green and blue. Since R = 6, the red neighbourhood of v can contain at most 5 vertices. Similarly, the green and blue neighbourhoods of v can contain at most 5 vertices each. Since every vertex, except for v itself, is in one of the red, green or blue neighbourhoods of v, the entire complete graph can have at most 1 + 5 + 5 + 5 = 16 vertices. Thus, we have R ≤ 17.
To see that R ≥ 17, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on K16, the so-called untwisted and twisted colourings. Both colourings are shown in the figures to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom.
If we select any colour of either the untwisted or twisted colouring on K16, and consider the graph whose edges are precisely those edges that have the specified colour, we will get the Clebsch graph.
It is known that there are exactly two edge colourings with 3 colours on K15 that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on K16, respectively.
It is also known that there are exactly 115 edge colourings with 3 colours on K14 that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours 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 such that if the hyperedges of a complete m-hypergraph of order R 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.

Infinite Ramsey theorem

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.
Proof: 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 a finite number of sets, then 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 c-colouring of the -element subsets of X, let a0 be an element of X and let Y = X \ . We then induce a c-colouring of the r-element subsets of Y, by just adding a0 to each r-element subset. By the induction hypothesis, there exists an infinite subset Y1 of Y such that every r-element subset of Y1 is coloured the same colour in the induced colouring. Thus there is an element a0 and an infinite subset Y1 such that all the -element subsets of X consisting of a0 and r elements of Y1 have the same colour. By the same argument, there is an element a1 in Y1 and an infinite subset Y2 of Y1 with the same properties. Inductively, we obtain a sequence such that the colour of each -element subset , ai, …, ai with i < i <... < i depends only on the value of i. Further, there are infinitely many values of i such that this colour will be the same. Take these ai's to get the desired monochromatic set.
A stronger but unbalanced infinite form of Ramsey's theorem for graphs, the Erdős–Dushnik–Miller theorem, states that every infinite graph contains either a countably infinite independent set, or an infinite clique of the same cardinality as the original graph.
In reverse mathematics, there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs and for infinite multigraphs. The multigraph version of the theorem is equivalent in strength to the arithmetical comprehension axiom, making it part of the subsystem ACA0 of second-order arithmetic, one of the big five subsystems in reverse mathematics. In contrast, by a theorem of David Seetapun, the graph version of the theorem is weaker than ACA0, and it does not fall into one of the big five subsystems.

Infinite version implies the finite

It is possible to deduce the finite Ramsey theorem from the infinite version by a proof by contradiction. Suppose the finite Ramsey theorem is false. Then there exist integers c, n, T such that for every integer k, there exists a c-colouring of without a monochromatic set of size T. Let Ck denote the c-colourings of without a monochromatic set of size T.
For any k, the restriction of a colouring in Ck+1 to is 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, neither is.
Similarly, the restriction of any colouring in is in, allowing one to define as the set of all such restrictions, a non-empty set. Continuing so, define for all integers m, k.
Now, for any integer k,, and each set is non-empty. Furthermore, Ck is finite as. It follows that the intersection of all of these sets is non-empty, and let. Then every colouring in Dk is the restriction of a colouring in Dk+1. Therefore, by unrestricting a colouring in Dk to a colouring in Dk+1, and continuing doing so, one constructs a colouring of without any monochromatic set of size T. This contradicts the infinite Ramsey theorem.
If a suitable topological viewpoint is taken, this argument becomes a standard compactness argument showing that the infinite version of the theorem implies the finite version.

Directed graph Ramsey numbers

It is also possible to define Ramsey numbers for directed graphs; these were introduced by. Let R be the smallest number Q such that any complete graph with singly directed arcs and with ≥ Q nodes contains an acyclic n-node subtournament.
This is the directed-graph analogue of what has been called R, 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.
We have R = 0, R = 1, R = 2, R = 4, R = 8, R = 14, R = 28, and 32 ≤ R ≤ 55.

Ramsey computation and quantum computers

Ramsey numbers can be determined by some universal quantum computers. The decision question is solved by determining whether the probe qubit exhibits resonance dynamics.