In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers.[1] 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 vertices or a red clique on vertices. (Here signifies an integer that depends on both and .)
Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by Frank 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,, and any given integers, there is a number,, such that if the edges of a complete graph of order are coloured with different colours, then for some between 1 and, it must contain a complete subgraph of order whose edges are all colour . The special case above has (and and).
Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex, . There are 5 edges incident to 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 the vertex,, to vertices,, and, are blue. (If not, exchange red and blue in what follows.) 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 contains a monochromatic, and therefore . 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,,,, such that the edge,, is red and the edge,, is blue. Firstly, any given vertex will be the middle of either (all edges from the vertex are the same colour), (four are the same colour, one is the other colour), or (three are the same colour, two are the other colour) such triples. Therefore, there are at most 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 are monochromatic.
Conversely, it is possible to 2-colour a without creating any monochromatic, showing that . The unique colouring is shown to the right. Thus .
The task of proving that was one of the problems of William Lowell Putnam Mathematical Competition in 1953, as well as in the Hungarian Math Olympiad in 1947.
A multicolour Ramsey number is a Ramsey number using 3 or more colours. There are (up to symmetries) only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely and .
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 . Consider the set of vertices that have a red edge to the vertex . This is called the red neighbourhood of . The red neighbourhood of 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 . Thus, the induced edge colouring on the red neighbourhood of has edges coloured with only two colours, namely green and blue. Since, the red neighbourhood of can contain at most 5 vertices. Similarly, the green and blue neighbourhoods of can contain at most 5 vertices each. Since every vertex, except for itself, is in one of the red, green or blue neighbourhoods of, the entire complete graph can have at most vertices. Thus, we have .
To see that, 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, the so-called untwisted and twisted colourings. Both colourings are shown in the figures to the right, with the untwisted colouring on the left, and the twisted colouring on the right.
If we select any colour of either the untwisted or twisted colouring on, 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 that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on, respectively.
It is also known that there are exactly 115 edge colourings with 3 colours on that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.
The theorem for the 2-colour case can be proved by induction on .[2] 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.
Lemma 1.
R(r,s)\leqR(r-1,s)+R(r,s-1).
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 edge is blue, and is in if is red. Because the graph has
R(r-1,s)+R(r,s-1)=|M|+|N|+1
|M|\geqR(r-1,s)
|N|\geqR(r,s-1).
M\cup\{v\}
In this 2-colour case, if and are both even, the induction inequality can be strengthened to:[3]
R(r,s)\leqR(r-1,s)+R(r,s-1)-1.
| t | |
| style\sum | |
| i=1 |
di
|M|=d1
|N|=t-1-d1
|M|\geqp-1,
|N|\geqq.
|M|
|M|\geqp
|N|\geqq.
|M|\geqp=R(r-1,s).
|N|\geqq=R(r,s-1)
Lemma 2. If, then
R(n1,...,nc)\leqR(n1,...,nc-2,R(nc-1,nc)).
Proof. Consider a complete graph of
R(n1,...,nc-2,R(nc-1,nc))
R(n1,...,nc-2,R(nc-1,nc)),
Lemma 1 implies that any is finite. The right hand side of the inequality in Lemma 2 expresses a Ramsey number for colours in terms of Ramsey numbers for fewer colours. Therefore, any is finite for any number of colours. This proves the theorem.
The numbers in Ramsey's theorem (and their extensions to more than two colours) 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. (The first exponential lower bound was obtained by Paul Erdős using the probabilistic method.) 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.[4] 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 (for colours) if brute force is used.[5] Therefore, the complexity for searching all possible graphs (via brute force) is for colourings and at most nodes.
The situation is unlikely to improve with the advent of quantum computers. One of the best-known searching algorithms for unstructured datasets exhibits only a quadratic speedup (c.f. Grover's algorithm) relative to classical computers, so that the computation time is still exponential in the number of nodes.[6] [7]
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 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.)
Using induction inequalities and the handshaking lemma, it can be concluded that, and therefore . There are only two graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among different 2-colourings of 16-node graphs, and only one graph (the Paley graph of order 17) among colourings. (This was proven by Evans, Pulham and Sheehan in 1979.) It follows that .
The fact that was first established by Brendan McKay and Stanisław Radziszowski in 1995.[8]
The exact value of is unknown, although it is known to lie between 43 (Geoffrey Exoo (1989)[9]) and 48 (Angeltveit and McKay (2017)[10]), inclusive.
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.[11]
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, by Stanisław Radziszowski, which is periodically updated.[12] [13] Where not cited otherwise, entries in the table below are taken from the January 2021 edition. (Note there is a trivial symmetry across the diagonal since .)
The inequality may be applied inductively to prove that
R(r,s)\leq\binom{r+s-2}{r-1}.
In particular, this result, due to Erdős and Szekeres, implies that when,
R(s,s)\leq(1+o(1))
| 4s | |
| \sqrt{\pis |
An exponential lower bound,
R(s,s)\geq(1+o(1))
| s | |
| \sqrt{2 |
e}2s/2,
was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is a huge gap between these two bounds: for example, for, this gives . Nevertheless, the exponential growth factors of either bound were not improved for a long time, and for the lower bound it still stands at . There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers are
[1+o(1)]
| \sqrt{2 | |
| s}{e} |
| ||||
| 2 |
\leqR(s,s)\leqs-(c4s,
due to Spencer and Conlon, respectively; a 2023 preprint by Campos, Griffiths, Morris and Sahasrabudhe claims to have made exponential progress using an algorithmic construction relying on a graph structure called a "book",[14] [15] improving the upper bound to
R(s,s)\leq(4-\varepsilon)sandR(s,t)\leqe-\delta\binom{s+t}{t}.
\varepsilon=2-7
\delta=50-1
\varepsilon
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
\Theta\left(\sqrt{nlogn}\right).
The upper bound for is given by Ajtai, Komlós, and Szemerédi,[16] the lower bound was obtained originally by Kim, and the implicit constant was improved independently by Fiz Pontiveros, Griffiths and Morris,[17] and Bohman and Keevash,[18] by analysing the triangle-free process. Furthermore, studying the more general "-free process" has set the best known asymptotic lower bounds for general off-diagonal Ramsey numbers,[19]
c's
| |||||||||||||
|
\leqR(s,t)\leqcs
| ts | |
| (logt)s |
.
For
s=4
c's
| ||||
| t |
(logt)-2\leqR(4,t)\leqcst3(logt)-2
c'st3(logt)-4
c'st3(logt)-d
There is a less well-known yet interesting analogue of Ramsey's theorem for induced subgraphs. Roughly speaking, instead of finding a monochromatic subgraph, we are now required to find a monochromatic induced subgraph. In this variant, it is no longer sufficient to restrict our focus to complete graphs, since the existence of a complete subgraph does not imply the existence of an induced subgraph. The qualitative statement of the theorem in the next section was first proven independently by Erdős, Hajnal and Pósa, Deuber and Rödl in the 1970s.[23] [24] [25] Since then, there has been much research in obtaining good bounds for induced Ramsey numbers.
Let be a graph on vertices. Then, there exists a graph such that any coloring of the edges of using two colors contains a monochromatic induced copy of (i.e. an induced subgraph of such that it is isomorphic to and its edges are monochromatic). The smallest possible number of vertices of is the induced Ramsey number .
Sometimes, we also consider the asymmetric version of the problem. We define to be the smallest possible number of vertices of a graph such that every coloring of the edges of using only red or blue contains a red induced subgraph of or blue induced subgraph of .
Similar to Ramsey's theorem, it is unclear a priori whether induced Ramsey numbers exist for every graph . In the early 1970s, Erdős, Hajnal and Pósa, Deuber and Rödl independently proved that this is the case. However, the original proofs gave terrible bounds (e.g. towers of twos) on the induced Ramsey numbers. It is interesting to ask if better bounds can be achieved. In 1974, Paul Erdős conjectured that there exists a constant such that every graph on vertices satisfies .[26] If this conjecture is true, it would be optimal up to the constant because the complete graph achieves a lower bound of this form (in fact, it's the same as Ramsey numbers). However, this conjecture is still open as of now.
In 1984, Erdős and Hajnal claimed that they proved the bound[27]
rind(H)\le
| |||||
| 2 |
.
Kohayakawa, Prömel and Rödl's bound remained the best general bound for a decade. In 2008, Fox and Sudakov provided an explicit construction for induced Ramsey numbers with the same bound.[29] In fact, they showed that every -graph with small and suitable contains an induced monochromatic copy of any graph on vertices in any coloring of edges of in two colors. In particular, for some constant, the Paley graph on vertices is such that all of its edge colorings in two colors contain an induced monochromatic copy of every -vertex graph.
In 2010, Conlon, Fox and Sudakov were able to improve the bound to, which remains the current best upper bound for general induced Ramsey numbers.[30] Similar to the previous work in 2008, they showed that every -graph with small and edge density contains an induced monochromatic copy of every graph on vertices in any edge coloring in two colors. Currently, Erdős's conjecture that remains open and is one of the important problems in extremal graph theory.
For lower bounds, not much is known in general except for the fact that induced Ramsey numbers must be at least the corresponding Ramsey numbers. Some lower bounds have been obtained for some special cases (see Special Cases).
While the general bounds for the induced Ramsey numbers are exponential in the size of the graph, the behaviour is much different on special classes of graphs (in particular, sparse ones). Many of these classes have induced Ramsey numbers polynomial in the number of vertices.
If is a cycle, path or star on vertices, it is known that is linear in .
If is a tree on vertices, it is known that .[31] It is also known that is superlinear (i.e.). Note that this is in contrast to the usual Ramsey numbers, where the Burr–Erdős conjecture (now proven) tells us that is linear (since trees are 1-degenerate).
For graphs with number of vertices and bounded degree, it was conjectured that, for some constant depending only on . This result was first proven by Łuczak and Rödl in 1996, with growing as a tower of twos with height .[32] More reasonable bounds for were obtained since then. In 2013, Conlon, Fox and Zhao showed using a counting lemma for sparse pseudorandom graphs that, where the exponent is best possible up to constant factors.[33]
Similar to Ramsey numbers, we can generalize the notion of induced Ramsey numbers to hypergraphs and multicolor settings.
We can also generalize the induced Ramsey's theorem to a multicolor setting. For graphs, define to be the minimum number of vertices in a graph such that any coloring of the edges of into colors contain an induced subgraph isomorphic to where all edges are colored in the -th color for some . Let (copies of).
It is possible to derive a bound on which is approximately a tower of two of height by iteratively applying the bound on the two-color case. The current best known bound is due to Fox and Sudakov, which achieves, where is the number of vertices of and is a constant depending only on .[34]
We can extend the definition of induced Ramsey numbers to -uniform hypergraphs by simply changing the word graph in the statement to hypergraph. Furthermore, we can define the multicolor version of induced Ramsey numbers in the same way as the previous subsection.
Let be a -uniform hypergraph with vertices. Define the tower function by letting and for, . Using the hypergraph container method, Conlon, Dellamonica, La Fleur, Rödl and Schacht were able to show that for, for some constant depending on only and . In particular, this result mirrors the best known bound for the usual Ramsey number when .[35]
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.[36]
Theorem. Let
X
[X]n
X
n
c
M
X
n
M
Proof: The proof is by induction on, the size of the subsets. For, 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, we prove it for . Given a -colouring of the -element subsets of, let be an element of and let
R(5,5)\leq48