Ramsey's theorem
Encyclopedia
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 2 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. Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds.
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(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).
In the following example, the formula R(3,3) provides a solution to the question which asks the minimum number of vertices a graph must contain in order to ensure that either (1) at least 3 vertices in the graph are connected or (2) at least 3 vertices in the graph are unconnected. Note that due to the symmetrical nature of the problem space, R(r,s) results in the same solution as R(s,r). This is not immediately evident in the example R(3,3) since the values of r and s are the same.
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 an entirely red triangle. Since this argument works for any colouring, any K6 contains a monochromatic K3, and therefore R(3,3) ≤ 6. The popular version of this is called the theorem on friends and strangers
.
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 (all edges from the vertex are the same color), 1×4=4 (four are the same color, one is the other color), or 2×3 = 6 (three are the same color, two are the other color) 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 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(3,3) > 5. The unique colouring is shown to the right. Thus R(3,3) = 6.
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.
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 partition the remaining vertices into two sets M and N, such that for every vertex w, w is in M if (v, w) is blue, and w is in N if (v, w) is red.
Because the graph has R(r - 1, s) + R(r, s - 1) = |M| + |N| + 1 vertices, it follows that either |M| ≥ R(r − 1, s) or |N| ≥ R(r, s − 1). 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
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 colours and 2 colours, and therefore exists and is a 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.
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
.
The complexity for searching all possible graphs is O
(2(n-1)(n-2)/2) for an upper bound of n nodes. http://www.learner.org/channel/courses/mathilluminated/units/2/textbook/06.php
Even the exact value of R(5,5) is unknown, although it is known to lie between 43 (Geoffrey Exoo) and 49
(Brendan McKay
and Stanisław Radziszowski) (inclusive). Using contemporary computational technology, it will take more than 2.144 * 10^250 years to exhaust all possible 2-colouring of graph of size 43.http://www.ezran.org/blog/2011/10/the-search-space-of-ramsey55/ Barring a breakthrough in theory, it is probable that the exact value of R(6,6) will remain unknown forever.
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
).
There is a trivial symmetry across the diagonal.
This table is extracted from a larger table compiled by Stanisław Radziszowski http://www.combinatorics.org/Surveys/index.html.
In particular, this result, due to Erdős
and Szekeres
, implies that when r = s,
An exponential lower bound,
was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. To this day, 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 R(3,t), it is known that they are of order
; this may be stated equivalently as saying that the smallest possible independence number
in an n-vertex triangle-free graph
is
. The upper bound for R(3,t) is given by Ajtai
, Komlós
, and Szemerédi
, the lower bound by Kim
. More generally, for off-diagonal Ramsey numbers R(s, t) with s fixed and t growing, the best known bounds are
due to Bohman and Keevash and Ajtai
, Komlós
and Szemerédi
respectively.
A multicolour Ramsey number is a Ramsey number using 3 or more colours. There is only one nontrivial multicolour Ramsey number for which the exact value is known, namely R(3,3,3) = 17.
Suppose that you have an edge colouring of a complete graph using 3 colours, red, yellow and green. Suppose further that the edge colouring 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 colouring on the green neighborhood of v has edges coloured with only two colours, 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 colouring on the complete graph on 16 vertices with 3 colours, which 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 figure to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom. In both colourings 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 colour of either the untwisted or twisted colouring on K16, and consider the graph whose edges are precisely those edges which have the specified colour, you will get the Clebsch graph
.
It is known that there are exactly two edge colourings with 3 colours on K15 which 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 which avoid monochromatic triangles, provided that we consider edge colourings which differ by a permutation of the colours as being the same.
s. 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.
terminology.
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.
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 (complete) 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 X, let a0 be an element of X and let Y = X\{a0}. 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 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 (r+1)-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 {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.
such that for every integer 
, there exists a 
-colouring of 
without a monochromatic set of size 
. Let 
denote the 
-colourings of 
without a monochromatic set of size 
.
For any k, the restriction of a colouring in
to 
(by ignoring the colour of all sets containing 
) is a colouring in 
. Define 
to be the colourings in 
which are restrictions of colourings in 
. Since 
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 
.
Now, for any integer
, 
, and each set is non-empty. Furthermore, 
is finite as 
. It follows that the intersection of all of these sets is non-empty, and let 
. Then every colouring in 
is the restriction of a colouring in 
. Therefore, by unrestricting a colouring in 
to a colouring in 
, and continuing doing so, one constructs a colouring of 
without any monochromatic set of size 
. 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.
(These were introduced by P. Erdős
& 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 colours is the two directions of the arcs,
the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
, one will find monochromatic complete subgraphs. For 2 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
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
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. It is understood to represent the smallest integer for which the theorem holds.
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
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
, 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(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
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
Without loss of generality
Without loss of generality is a frequently used expression in mathematics...
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 an entirely red triangle. Since this argument works for any colouring, any K6 contains a monochromatic K3, and therefore R(3,3) ≤ 6. The popular version of this is called the theorem on friends and strangers
Theorem on friends and strangers
The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory.-Statement:Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met...
.
An alternate proof works by double counting
Double counting (proof technique)
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set...
. 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 (all edges from the vertex are the same color), 1×4=4 (four are the same color, one is the other color), or 2×3 = 6 (three are the same color, two are the other color) 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 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(3,3) > 5. The unique colouring is shown to the right. Thus R(3,3) = 6.
Proof of the theorem
First we prove the theorem for the 2-colour case, by inductionMathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
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.
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 partition the remaining vertices into two sets M and N, such that for every vertex w, w is in M if (v, w) is blue, and w is in N if (v, w) is red.
Because the graph has R(r - 1, s) + R(r, s - 1) = |M| + |N| + 1 vertices, it follows that either |M| ≥ R(r − 1, s) or |N| ≥ R(r, s − 1). 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
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 colours and 2 colours, and therefore exists and is a 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 obtained by Paul ErdősPaul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
using the probabilistic method
Probabilistic method
The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the...
.) 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
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
.
The complexity for searching all possible graphs is O
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
(2(n-1)(n-2)/2) for an upper bound of n nodes. http://www.learner.org/channel/courses/mathilluminated/units/2/textbook/06.php
Even the exact value of R(5,5) is unknown, although it is known to lie between 43 (Geoffrey Exoo) and 49
(Brendan McKay
Brendan McKay
Brendan Damien McKay is a Professor in the Research School of Computer Science at the Australian National University . He has published extensively in combinatorics....
and Stanisław Radziszowski) (inclusive). Using contemporary computational technology, it will take more than 2.144 * 10^250 years to exhaust all possible 2-colouring of graph of size 43.http://www.ezran.org/blog/2011/10/the-search-space-of-ramsey55/ Barring a breakthrough in theory, it is probable that the exact value of R(6,6) will remain unknown forever.
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
Brendan McKay
Brendan Damien McKay is a Professor in the Research School of Computer Science at the Australian National University . He has published extensively in combinatorics....
).
| 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 | 73–115 | 92–149 |
| 5 | 1 | 5 | 14 | 25 | 43–49 | 58–87 | 80–143 | 101–216 | 126–316 | 144–442 |
| 6 | 1 | 6 | 18 | 35–41 | 58–87 | 102–165 | 113–298 | 132–495 | 169–780 | 179–1171 |
| 7 | 1 | 7 | 23 | 49–61 | 80–143 | 113–298 | 205–540 | 217–1031 | 241–1713 | 289–2826 |
| 8 | 1 | 8 | 28 | 56–84 | 101–216 | 132–495 | 217–1031 | 282–1870 | 317–3583 | 331-6090 |
| 9 | 1 | 9 | 36 | 73–115 | 126–316 | 169–780 | 241–1713 | 317–3583 | 565–6588 | 581–12677 |
| 10 | 1 | 10 | 40–43 | 92–149 | 144–442 | 179–1171 | 289–2826 | 331-6090 | 581–12677 | 798–23556 |
There is a trivial symmetry across the diagonal.
This table is extracted from a larger table compiled by Stanisław Radziszowski http://www.combinatorics.org/Surveys/index.html.
Asymptotics
The inequality R(r, s) ≤ R(r − 1, s) + R(r, s − 1) may be applied inductively to prove thatIn particular, this result, due to Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
and Szekeres
George Szekeres
George Szekeres AM was a Hungarian-Australian mathematician.-Early years:Szekeres was born in Budapest, Hungary as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked six years in Budapest as an analytical chemist. He married Esther Klein in 1936...
, implies that when r = s,
An exponential lower bound,
was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. To this day, 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
Joel Spencer
Joel Spencer is an American mathematician. He is a combinatorialist who has worked on probabilistic methods in combinatorics and on Ramsey theory. He received his doctorate from Harvard University in 1970, under the supervision of Andrew Gleason...
and Conlon respectively.
For the off-diagonal Ramsey numbers R(3,t), it is known that they are of order
; this may be stated equivalently as saying that the smallest possible independence numberIndependent set (graph theory)
In graph theory, an independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set I of vertices such that for every two vertices in I, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in I...
in an n-vertex triangle-free graph
Triangle-free graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally...
is
. The upper bound for R(3,t) is given by AjtaiMiklós Ajtai
Miklós Ajtai is a computer scientist at the IBM Almaden Research Center. In 2003 he received the Knuth Prize for his numerous contributions to the field, including a classic sorting network algorithm Miklós Ajtai (born 2 July 1946, Budapest, Hungary) is a computer scientist at the IBM Almaden...
, Komlós
János Komlós (mathematician)
János Komlós is a Hungarian-American mathematician, working in probability theory and discrete mathematics. He is a professor of mathematics at Rutgers University since 1988. He graduated from the Eötvös Loránd University, then became a fellow at the Mathematical Institute of the Hungarian Academy...
, and Szemerédi
Endre Szemerédi
Endre Szemerédi is a Hungarian mathematician, working in the field of combinatorics and theoretical computer science. He is the State of New Jersey Professor of computer science at Rutgers University since 1986...
, the lower bound by Kim
Jeong Han Kim
Jeong Han Kim is a South Korean mathematician specializing in combinatorics and computational mathematics. He studied physics and mathematical physics at Yonsei University, and earned his Ph.D in mathematics at Rutgers University...
. More generally, for off-diagonal Ramsey numbers R(s, t) with s fixed and t growing, the best known bounds are
due to Bohman and Keevash and Ajtai
Miklós Ajtai
Miklós Ajtai is a computer scientist at the IBM Almaden Research Center. In 2003 he received the Knuth Prize for his numerous contributions to the field, including a classic sorting network algorithm Miklós Ajtai (born 2 July 1946, Budapest, Hungary) is a computer scientist at the IBM Almaden...
, Komlós
János Komlós (mathematician)
János Komlós is a Hungarian-American mathematician, working in probability theory and discrete mathematics. He is a professor of mathematics at Rutgers University since 1988. He graduated from the Eötvös Loránd University, then became a fellow at the Mathematical Institute of the Hungarian Academy...
and Szemerédi
Endre Szemerédi
Endre Szemerédi is a Hungarian mathematician, working in the field of combinatorics and theoretical computer science. He is the State of New Jersey Professor of computer science at Rutgers University since 1986...
respectively.
A Multicolour Example: R(3,3,3) = 17
Suppose that you have an edge colouring of a complete graph using 3 colours, red, yellow and green. Suppose further that the edge colouring 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 colouring on the green neighborhood of v has edges coloured with only two colours, 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 colouring on the complete graph on 16 vertices with 3 colours, which 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 figure to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom. In both colourings 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.
Clebsch graph
In the mathematical field of graph theory, the Clebsch graph is an undirected graph with 16 vertices and 40 edges. It is named after Alfred Clebsch, a German mathematician who discovered it in 1868...
.
It is known that there are exactly two edge colourings with 3 colours on K15 which 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 which avoid monochromatic triangles, provided that we consider edge colourings which differ by a permutation of the colours as being the same.
Extensions of the theorem
The theorem can also be extended to hypergraphHypergraph
In mathematics, a hypergraph is a generalization of a graph, where an edge can connect any number of vertices. Formally, a hypergraph H is a pair H = where X is a set of elements, called nodes or vertices, and E is a set of non-empty subsets of X called hyperedges or links...
s. 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.
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-theoreticSet theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
terminology.
Theorem: Let X be some countably
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
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.
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 (complete) 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 X, let a0 be an element of X and let Y = X\{a0}. 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 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 (r+1)-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 {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.
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 such that for every integer , there exists a -colouring of without a monochromatic set of size . Let denote the -colourings of without a monochromatic set of size .For any k, the restriction of a colouring in
to (by ignoring the colour of all sets containing ) is a colouring in . Define to be the colourings in which are restrictions of colourings in . Since 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 .Now, for any integer
, , and each set is non-empty. Furthermore, is finite as . It follows that the intersection of all of these sets is non-empty, and let . Then every colouring in is the restriction of a colouring in . Therefore, by unrestricting a colouring in to a colouring in , and continuing doing so, one constructs a colouring of without any monochromatic set of size . This contradicts the infinite Ramsey theorem.If a suitable topological viewpoint is taken, this argument becomes a standard compactness argument
Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...
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 P. Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
& 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 colours is the two directions of the arcs,
the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")
See also
- Paris–Harrington theoremParis–Harrington theoremIn mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory is true, but not provable in Peano arithmetic...
- Sim (pencil game)Sim (pencil game)The game of Sim is played by two players on a board consisting of six dots . Each dot is connected to every other dot by a line.Two players take turns coloring any uncolored lines...
- Infinite Ramsey theory
- Van der Waerden number
External links
- Ramsey@Home is a distributed computingDistributed computingDistributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...
project designed to find new lower bounds for various Ramsey numbers using a host of different techniques. - Radziszowski's survey of small Ramsey numbers
- this lisp code computes upper bounds for particular Ramsey numbers given the general rules in Radziszowski
- Survey of directed-graph Ramsey numbers
- Ramsey Number - from MathWorld (contains lower and upper bounds up to R(19,19))
- Ramsey Number - Geoffrey Exoo (Contains R(5,5) > 42 counter-proof)
- A proof that R(3,6) = 18