《离散概率方法》研究生课程参考资料（书籍文献）Ramsey's Theorem（Wikipedia）.pdf_大学文库

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 the vertex,v,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,() (s,()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 Ke contains a monochromatic K3,and therefore R(3,3)s 6.The popular version of this is called the theorem on friends and strangers. A2-6 An altemative 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 0x 5=0(all edges from the vertex are the same colour),1 x 4=4(four are the same colour,one is the other colour),or 2 x 3=6(three are the same colour,two are the other colour)such triples.Therefore,there are at most 6x 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 Ks without creating any monochromatic K3.showing that R(3,3)>5. The uniquel21 colouring is shown to the right.Thus R(3,3)=6. The task of proving that R(3,3)s6 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 r+s.3]It is clear from the definition that for all n,R(n,2)=R(2,n)=n.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. Lemma1.Rr,s）≤R(r-1,s+R(r,s-1) of(1+R(-1)vertice whose edges ed with N.suc w lours.Pick a ver R(revery ve .wi s in Mif ap"w is blu W) th )+R(r, and w rIM≥ R(r e grapl M≥R(, -1).In the former case,if M a red Ks then s does the original graph nd we are finished Otherwise M has a blue Kr-1 and so M U {v)has a blue Kr by the definition of M.The latter case is analogous.Thus the claim is true and we have completed the proof for 2 colours. In this 2-colocase if R(r,s)and R(r,s-1)are both even,the induction inequality can be strengthened to: R(r,s)s R(r-1,s)+R(r,s-1)-1

A 2-edge-labeling 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 the vertex, v, 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 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, (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 colour), 1 × 4 = 4 (four are the same colour, one is the other colour), or 2 × 3 = 6 (three are the same colour, two are the other colour) 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[2] colouring is shown to the right. Thus R(3, 3) = 6. The task of proving that R(3, 3) ≤ 6 was one of the problems of William Lowell Putnam Mathematical Competition in 1953, as well as in the Hungarian Math Olympiad in 1947. The theorem for the 2-colour case, can be proved by induction on r + s. [3] It is clear from the definition that for all n, R(n, 2) = R(2, n) = n. 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. Lemma 1. R(r, s) ≤ R(r − 1, s) + R(r, s − 1): Proof. Consider a complete graph on R(r − 1, s) + R(r, s − 1) vertices whose edges are coloured with two colours. 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 M ∪ {v} has a blue Kr by the definition of M. 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 R(r − 1, s) and R(r, s − 1) are both even, the induction inequality can be strengthened to:[4] R(r, s) ≤ R(r − 1, s) + R(r, s − 1) − 1. Proof of the theorem 2-colour case

Proof.Suppose p R(r-1,s)and g=R(r,s-1)are both even.Let t=p+q-1 and consider a two-coloured graph of t vertices.If di is degree of i-th vertex in the graph,then,according to the Handshaking lemma,is even.Given that tis odd,there must be an evenAssume dis even,M and Nare the vertices incident to vertex 1 in the blue and red subgraphs,respectively.Then both M=d and N=t-1-d are even.According to the Pigeonhole principle,either M>p-1,or N >g.Since M]is even,while p-1 is odd,the first inequality can be strengthened,so either M2p or N2g =R(r-1,).Then either the M subgraph has a red K and the p lete,or has a blue K-1 which along with vertex 1 makes a blue Kr.The case=R(r,s-1)is treated similarly. Case of more colours Lemma 2.If c>2,then R(m...n)s R(n..n.R(nc-1.nc)). Proof.Consider a complete graph of R(n n-2,R(n-,n)vertices and colour its edges with c colours and are th olou Thus the 1) chromatically coloured with colour ifor some 1sisc-2or aKg cooured in the blurred colourIn the former cas ecover our sight again and seefrom the definitionof either a(c 1)-mon chrome either case the proo is complete. Lemma 1 impli that any R(rs)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(n,...,n)is finite for any number of colours.This proves the theorem. Ramsey numbers The numbers R(r,s)in Ramsey's theorem (and their extensions to more than two colours)are known as Ramsey numbers.The ramsey number.R(m.n).gives the solution to the party problem.which asks the minimum number of guests.R(m,n).that must be invited so that at least m will know each other or at least n will not know each other.In the language of graph theory,the Ramsey number is the minimum number of vertices,v=R(m,n),such that all undirected simple graphs of order v,contain a clique of order m,or an independent set of order n.Ramsey's theorem states that such a number exists for all m and n. By symmetry,it is true that R(m,n)=R(n,m).An upper bound for R(r,s)can be extracted from the proof of the theorem,and other arguments give lower bounds.(The first exponential lower bound was obtained by Paul Erdos 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 r and s for which we know the exact value of R(r.s). Computing a lower bound L forR(s) usually requires exhibiting a blue/red colouring of the graph KL-1 with no blue Kr subgraph and no red Ks subgraph.Such a counterexample is called a Ramsey graph. Brendan McKay maintains a list of known Ramsey graphs.[5]Upp oer bounds are often considerably more difficult to establish one gither has to check all nossible co olourings to confirm the absence of a counterexample,or to present a mathematical argument for its absence.A sophisticated c ter program does not need to look at all colourings individually in order to eliminate all of them:nevertheless it is a verv difficult computational task that existing software can only manage on small sizes.Each complete graph K

Proof. Suppose p = R(r − 1, s) and q = R(r, s − 1) are both even. Let t = p + q − 1 and consider a two-coloured graph of t vertices. If is degree of -th vertex in the graph, then, according to the Handshaking lemma, is even. Given that t is odd, there must be an even . Assume is even, M and N 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 M 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. Lemma 2. If c>2, then R(n1 , …, nc ) ≤ R(n1 , …, nc−2 , R(nc−1 , nc )). Proof. Consider a complete graph of R(n1 , …, nc−2 , R(nc−1 , nc )) 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. Due to the definition of R(n1 , …, nc−2 , R(nc−1 , nc )), such a graph 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. Lemma 1 implies that any R(r,s) 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(n1 , …, nc ) is finite for any number of colours. This proves the theorem. The numbers R(r, s) in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. The Ramsey number, R(m, n), gives the solution to the party problem, which asks the minimum number of guests, R(m, n), that must be invited so that at least m will know each other or at least n will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices, v = R(m, n), such that all undirected simple graphs of order v, contain a clique of order m, or an independent set of order n. Ramsey's theorem states that such a number exists for all m and n. By symmetry, it is true that R(m, n) = R(n, m). An upper bound for R(r, s) 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 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. Such a counterexample is called a Ramsey graph. Brendan McKay maintains a list of known Ramsey graphs. [5] 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 Kn Case of more colours Ramsey numbers

has 1 2 n(n − 1) edges, so there would be a total of c n(n − 1) 2 graphs to search through (for c colours) if brute force is used.[6] Therefore, the complexity for searching all possible graphs (via brute force) is O(c n 2 ) for c colourings and an upper bound of n nodes. As described above, R(3, 3) = 6. It is easy to prove that R(4, 2) = 4, and, more generally, that R(s, 2) = s for all s: a graph on s − 1 nodes with all edges coloured red serves as a counterexample and proves that R(s, 2) ≥ s; among colourings of a graph on s nodes, the colouring with all edges coloured red contains a snode 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, it can be concluded that R(4, 3) ≤ R(4, 2) + R(3, 3) − 1 = 9, and therefore R(4, 4) ≤ R(4, 3) + R(3, 4) ≤ 18. There are only two (4, 4, 16) graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among 6.4 × 10 22 different 2-colourings of 16-node graphs, and only one (4, 4, 17) graph (the Paley graph of order 17) among 2.46 × 10 26 colourings. [5] (This was proven by Evans, Pulham and Sheehan in 1979.) It follows that R(4, 4) = 18. The fact that R(4, 5) = 25 was first established by Brendan McKay and Stanisław Radziszowski in 1995.[7] The exact value of R(5, 5) is unknown, although it is known to lie between 43 (Geoffrey Exoo (1989)[8] ) and 48 (Angeltveit and McKay (2017)[9] ) (inclusive). Erdős asks us to 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, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens. — Joel Spencer [10] In 1997, McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture that R(5, 5) = 43. They were able to construct exactly 656 (5, 5, 42) graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a (5, 5, 43) graph.[11] For R(r, s) with r, s > 5, only weak bounds are available. Lower bounds for R(6, 6) and R(8, 8) have not been improved since 1965 and 1972, respectively. [12] R(r, s) with r, s ≤ 10 are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. R(r, s) with r, s < 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 is the Dynamic Survey 1 of the Electronic Journal of Combinatorics. [12] 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 R(r, s) = R(s, r)

The only two 3-colourings of K16 with no monochromatic K3 . The untwisted colouring (above) and the twisted colouring (below). 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 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, 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 are (up to symmetries) only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely R(3, 3, 3) = 17 and R(3, 3, 4) = 30.[12] 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(3, 3) = 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(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 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. A multicolour example: R(3, 3, 3) = 17

It is known that there are exactly two edge colourings with 3 colours on Ki5 that avoid monochromatic triangles,which can be constructed avertexfrom the untwisted and twisted cooringo It is also known that there are exactly 115 edge colour colours on Ki that avoid monochromatic triangles.provided that we Extensions of the theorem Clebsch graph The theorem can also be extended to hypergraphs.An m-hypergraph is a graph who in a normal graph ar edge isa set of 2 verices.The l statement of Ramsey's theorem for hypergraphs is that for any integersm and c,and any integers n,...,nc there is an integer R(n,...ncic,m)such that if the hyperedges of a complete m-hypergraph of order R(n,...,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 n;whose hyperedges are all colour i This theorer n 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 commonl called Ramsey's th ontext finit often ore As int d by tatio area are usually phrased in set-theoretic temminology. Theorem let x he s nd colour the el ents of x(n)(the subsets of X of size in c differ The e exists some infinite subset M of x subsets of M all have the same colour. such that the size n 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 sr,we prove it for n =r+1.Given a c-colouring of the (r+1)-element subsets of X,let ao be an element of X and let Y=X\{ao).We then induce a c-colouring of the r-element subsets of Y.by just adding ao to each r-element subset (to get an (r+1)-element subset of x.By the induction hypothesis,there xists an infinite subset Y of Y such that ev ry r-element subset of Y is coloured the sam e colour in the induced colouring.Thus ther e is ar element o and an infinite subset h that all the (r+1)-element subsets of X consisting of ao and r elements of Y have the same colour.By the same argument,there is an element a in Y and an infinite subset Y2 of Y with the same properties.Inductively,we obtain a sequence ao.a2 ..such that the colour of each (r+1)-element subset (a 1ai2 with i(1)<i(2)<<i(r+1 )dep nds only on the value of i(1).Furtl are infinite of i(n) such that this olour will be the e Take thes e ain)'s to ge states that every infinite graph contains

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. 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. 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. [14] Theorem. Let X be some 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 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 (r + 1)-element subsets of X, let a0 be an element of X and let Y = X \ {a0}. We then induce a c-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 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 (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. 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.[15] Extensions of the theorem Infinite Ramsey theorem

In reverse mathematics,there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs(the case n=2)and for infinite multigraphs (the case n 23).The multigraph version of the theorem is equivalent in strength to the arithmetical comprehension axiom,making it part of the subsystem ACAo 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 ACAo,and (combining Seetapun's result with others)it does not fall into one of the big five subsystems.6 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 Cr denote the c-colourings of [( without a monochromatic set of size t. For any k,the restriction of a colouring in Ck+to[(by ignoring the colour of all sets containingk+1)is a colouring in Ci.Define C to be the colourings in Cr which are restrictions of colourings in C.Since Ck+is not empty,neither is C 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 Cm for all integers m,k. Now,for any integer k,C..,and each set is non-empty.Furthermore,Ck is finite as follows that the intersection of all of these sets is non-empty.and let D=C.Then every colouring in Dk is the restriction of a colouring in D+.Therefore, by unrestricting a colouring in D to a colouring in D,and continuing doing so,one constructs a colouring of N)without any monochromatic set of size T.This contradicts the infinite Ramsey theorem. If a suitable topologica vie that the infin ewp int is taken,this ent bo Directed graph Ramsey numbers It is also possible to define Ramsey numbers for directed graphs;these were introduced by P.Erdos and L. Moser(1964).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 subtoumament. graph analogue of what(above)has mber Z such that olete undirect node he a logue of the rs is the two directions of the arcs.the analogue of monochromatic"is"all arc-amows point the same wayi.e."acyclic. We have R(0)=0,R(1)=1,R(2)=2,R3)=4,R(4)=8,R5)=14,R(6)=28,and32≤R(7)≤54.18 Ramsey computation and quantum computers

In reverse mathematics, there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs (the case n = 2) and for infinite multigraphs (the case n ≥ 3). 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 (combining Seetapun's result with others) it does not fall into one of the big five subsystems. [16] 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 (by ignoring the colour of all sets containing k + 1) 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.[17] It is also possible to define Ramsey numbers for directed graphs; these were introduced by P. Erdős and L. Moser (1964). 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.") We have R(0) = 0, R(1) = 1, R(2) = 2, R(3) = 4, R(4) = 8, R(5) = 14, R(6) = 28, and 32 ≤ R(7) ≤ 54.[18] Ramsey numbers can be determined by some universal quantum computers. The decision question is solved by determining whether the probe qubit exhibits resonance dynamics. [19] Infinite version implies the finite Directed graph Ramsey numbers Ramsey computation and quantum computers

See also ·Ramsey cardinal Paris-Harrington theorem ·Sim(pencil game)) Infinite Ramsey theory Van der Waerden number ·Ramsey game Erdos -Rado theorem Notes 1.Some authors restrict the values to be greater than one.for example(Brualdi 2010)and(Harar 1972).thus avoiding a discussion of edge colouring a graph with no edges.while others rephrase the statement of the theorem to require,in a simple graph,either an r-clique or an s- independent set,see(Gross 2008)or (Erd6s Szekeres 1935).In this form,the consideration of graphs with one vertex is more natural. 2.up to automorphisms of the araph 3.D an(2006)."pa au/Publ/Gaze te/200 /Nov06/Nov06. p "Party Acquaintances"(http://www.cut-the-knot.org/Curriculum/Combinatorics/ThreeOrThree.sh tml#nequality2) 5."Ramsey Graphs"(http://cs.anu.edu.au/-bdm/data/ramsey.html). 6.2.6 Ramsey Theory from Mathematics Illuminated(http://www.learner.org/channel/courses/math illuminated/units/2/textbook/06.php) 7.Brendan D.McKay.Stanislaw P.Radziszowski(May 1995)."R(4,5)=25"(htp://users.cecs.anu edu au-hd s/rA5 df (PDE)loumal of Granh The /1gQ1200 doi10.1002jgt3190190304htps:1doi.0rg10.1002%2Fjgt3190190304). 8.Ex00 98.doi10.1002jgt3190130113ht lower bound for R(5,5) Journal of Graph Theory.13(1):97- doi.0rg/10.10029%2Fjgt3190130113). 9.Vigleik Angeltveit;Brendan McKay(2017)."R(5,5)<48".arXiv:1703.08768v2(https://arxiv.or g/abs/1703.08768v2)[math.CO (https://arxiv.org/archive/math.CO)]. 10.Joel H.Spencer(1994),Ten Lectures on the Probabilistic Method (https://archive.org/details/ten lecturesonpro0000spen/page/4),SIAM,p.4(https://archive.org/details/tenlecturesonpro0000sp en/page/4),ISBN978-0-89871-325-1 11 Brendan d.McKay.sStanistaw P Radziszowski (1997)"Subaraph Countina ldentities and Ramsey Numbers"(http://cs.anu.edu.au/-bdm/papers/r55.pdf)(PDF).Journal of Combinatorial The0y.69(2y:193-209.doi1:10.1006/jctb.1996.1741htps:ldoi.org/10.1006%2 Fjctb.1996.174 1) 12.Radziszowski,Stanislaw(2011)."Small Ramsey Numbers"(http://www.combinatorics.org/ojs/i ndex.php/eljclarticle/view/DS1/pdf).Dynamic Surveys.Electronic Journal of Combinatorics. doi:10.37236/21htps:ldo1.0rg/10.37236%2F21) 13 Exoo Geoffrev Tatarevic Milos (2015)"N ds for 28 classical Ramse Numbers"(htp:ww combinatorics prlisndppelicarticlewiew22i3cnic Journal of Combinatorics.22 (3):3.arXiv:1504.02403(https://arxiv.org/abs/1504.02403). Bibcode:2015arXiv150402403E(https://ui.adsabs.harvard.edu/abs/2015arXiv150402403E) doi:10.37236/5254htps:/doi.0rg/10.37236%2F5254) 14.Martin Gould."Ramsey Theory"(http://people.maths.ox.ac.uk/-gouldm/ramsey.pdf)(PDF)

Ramsey cardinal Paris–Harrington theorem Sim (pencil game) Infinite Ramsey theory Van der Waerden number Ramsey game Erdős–Rado theorem 1. Some authors restrict the values to be greater than one, for example (Brualdi 2010) and (Harary 1972), thus avoiding a discussion of edge colouring a graph with no edges, while others rephrase the statement of the theorem to require, in a simple graph, either an r-clique or an sindependent set, see (Gross 2008) or (Erdős & Szekeres 1935). In this form, the consideration of graphs with one vertex is more natural. 2. up to automorphisms of the graph 3. Do, Norman (2006). "Party problems and Ramsey theory" (http://www.austms.org.au/Publ/Gaze tte/2006/Nov06/Nov06.pdf#page=17) (PDF). Austr. Math. Soc. Gazette. 33 (5): 306–312. 4. "Party Acquaintances" (http://www.cut-the-knot.org/Curriculum/Combinatorics/ThreeOrThree.sh tml#inequality2). 5. "Ramsey Graphs" (http://cs.anu.edu.au/~bdm/data/ramsey.html). 6. 2.6 Ramsey Theory from Mathematics Illuminated (http://www.learner.org/channel/courses/math illuminated/units/2/textbook/06.php) 7. Brendan D. McKay, Stanislaw P. Radziszowski (May 1995). "R(4,5) = 25" (http://users.cecs.anu. edu.au/~bdm/papers/r45.pdf) (PDF). Journal of Graph Theory. 19 (3): 309–322. doi:10.1002/jgt.3190190304 (https://doi.org/10.1002%2Fjgt.3190190304). 8. Exoo, Geoffrey (March 1989). "A lower bound for R(5, 5)". Journal of Graph Theory. 13 (1): 97– 98. doi:10.1002/jgt.3190130113 (https://doi.org/10.1002%2Fjgt.3190130113). 9. Vigleik Angeltveit; Brendan McKay (2017). " ". arXiv:1703.08768v2 (https://arxiv.or g/abs/1703.08768v2) [math.CO (https://arxiv.org/archive/math.CO)]. 10. Joel H. Spencer (1994), Ten Lectures on the Probabilistic Method (https://archive.org/details/ten lecturesonpro0000spen/page/4), SIAM, p. 4 (https://archive.org/details/tenlecturesonpro0000sp en/page/4), ISBN 978-0-89871-325-1 11. Brendan D. McKay, Stanisław P. Radziszowski (1997). "Subgraph Counting Identities and Ramsey Numbers" (http://cs.anu.edu.au/~bdm/papers/r55.pdf) (PDF). Journal of Combinatorial Theory. 69 (2): 193–209. doi:10.1006/jctb.1996.1741 (https://doi.org/10.1006%2Fjctb.1996.174 1). 12. Radziszowski, Stanisław (2011). "Small Ramsey Numbers" (http://www.combinatorics.org/ojs/i ndex.php/eljc/article/view/DS1/pdf). Dynamic Surveys. Electronic Journal of Combinatorics. doi:10.37236/21 (https://doi.org/10.37236%2F21). 13. Exoo, Geoffrey; Tatarevic, Milos (2015). "New Lower Bounds for 28 Classical Ramsey Numbers" (http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p11). Electronic Journal of Combinatorics. 22 (3): 3. arXiv:1504.02403 (https://arxiv.org/abs/1504.02403). Bibcode:2015arXiv150402403E (https://ui.adsabs.harvard.edu/abs/2015arXiv150402403E). doi:10.37236/5254 (https://doi.org/10.37236%2F5254). 14. Martin Gould. "Ramsey Theory" (http://people.maths.ox.ac.uk/~gouldm/ramsey.pdf) (PDF). See also Notes

15.Dushnik,Ben;Miller,E.W.(1941)."Partially ordered sets".American Journal of Mathematics 63(3):600-610.doi:10.23072371374(htps:ldoi.org/10.2307%2F2371374) hdl:10338.dmlcz/100377(https://hdl.handle.net/10338.dmlcz%2F100377).JSTOR 2371374(htt 16.Hirschfeldt,Denis R.(2014).Slicing the Truth.Lecture Notes Series of the Institute for Mathematical Sciences,National University of Singapore.28.World Scientific. 17.Diestel,Reinhard(2010)."Chapter 8,Infinite Graphs".Graph Theory (4 ed.).Heidelberg Springer--Verlag.pp.209-2010.ISBN978-3-662-53621-6. 18.Smith,Warren D.:Exoo,Geoff,Partial Answer to Puzzle #27:A Ramsey-like quantity (http://Ra ngeVoting.org/PuzzRamsey.html).retrieved 2020-06-02 19 ang,He 20161"DE eg502301. s on abs1510.01884) Physical ttps:/ Bibcode:2016PhRvA..93c2301W (https://ui.adsabs.h org/ du/abs/2016PhRvA..93c2301W) doi10.1103/PhysRevA.93.032301htps:/doi.0rg/10.11039%2 FPhysRevA.93.032301. References Ajtai,Miklos:Koml6s,Janos:Szemeredi,Endre(1980)."A note on Ramsey numbers",J. C0mbin.Theory Ser..A,29(3):354-360,doi:10.1016/0097-3165(80)90030-8htps:ldoi.org/10 1016%2F0097-31659%28809%2990030-8). Tom;Keevash,Peter(2010),"The early evolution of the H-free process".Invent. Math,1812y:291-336,arXiv:0908.0429htps/axiv.org/abs/0908.0429). Bibcode:2010InMat.181..291B(https://ui.adsabs.harvard.edu/abs/2010InMat.181..291B) doi:10.1007/s00222-010-0247-×htps:doi1.0rg/10.1007%2Fs00222-010-0247-x) Brualdi,Richard A.(2010),Introductory Combinatorics(5th ed.),Prentice-Hall,pp.77-82 1SBN978.0-13-602040-0 ath/0607788 d ey num di104007 annals.2009.170.941G 0rg/10.40079%2 Fannals.2009.170.941. MR 2552114(https://www.ams.org/mathscinet-getitem?mr=2552114). arks on thethe 7-08785-1) Erd6s,P.;M r.L.(1964)."nh repre on of di ected s of orderings"(ht Mate atikai Kutato Inteze erenek Kozlemenyei,9:125-132.MR 0168494 (hups:/www.ams.org/ mathscinet-getitem?mr=0168494) Erdos,Pau do (PDE) natorial pr doi:10 100 7978.0-8176484283hps010r010100762F9780-8176484283. 1sBN978-0-8176-4841-1. ,5) :97-98 ://do /10.10 262Fj9t3190130113 Graham,R.;Rothschild,B.;Spencer,J.H.(1990),Ramsey Theory,New York:John Wiley and ons Gross,Jonathan L.(2008),Combinatorial Methods with Computer Applications,CRC Press, p.458,1SBN978-1-58488-743-0 Harary,Frank(1972),Graph Theory,Addison-Wesley,pp.16-17,ISBN 0-201-02787-9 s9%2Fs2-3

Ajtai, Miklós; Komlós, János; Szemerédi, Endre (1980), "A note on Ramsey numbers", J. Combin. Theory Ser. A, 29 (3): 354–360, doi:10.1016/0097-3165(80)90030-8 (https://doi.org/10. 1016%2F0097-3165%2880%2990030-8). Bohman, Tom; Keevash, Peter (2010), "The early evolution of the H-free process", Invent. Math., 181 (2): 291–336, arXiv:0908.0429 (https://arxiv.org/abs/0908.0429), Bibcode:2010InMat.181..291B (https://ui.adsabs.harvard.edu/abs/2010InMat.181..291B), doi:10.1007/s00222-010-0247-x (https://doi.org/10.1007%2Fs00222-010-0247-x) Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Prentice-Hall, pp. 77–82, ISBN 978-0-13-602040-0 Conlon, David (2009), "A new upper bound for diagonal Ramsey numbers", Annals of Mathematics, 170 (2): 941–960, arXiv:math/0607788v1 (https://arxiv.org/abs/math/0607788v1), doi:10.4007/annals.2009.170.941 (https://doi.org/10.4007%2Fannals.2009.170.941), MR 2552114 (https://www.ams.org/mathscinet-getitem?mr=2552114). Erdős, Paul (1947), "Some remarks on the theory of graphs", Bull. Amer. Math. Soc., 53 (4): 292–294, doi:10.1090/S0002-9904-1947-08785-1 (https://doi.org/10.1090%2FS0002-9904-194 7-08785-1). Erdős, P.; Moser, L. (1964), "On the representation of directed graphs as unions of orderings" (ht tps://users.renyi.hu/~p_erdos/1964-22.pdf) (PDF), A Magyar Tudományos Akadémia, Matematikai Kutató Intézetének Közleményei, 9: 125–132, MR 0168494 (https://www.ams.org/ mathscinet-getitem?mr=0168494) Erdős, Paul; Szekeres, George (1935), "A combinatorial problem in geometry" (http://www.num dam.org/article/CM_1935__2__463_0.pdf) (PDF), Compositio Mathematica, 2: 463–470, doi:10.1007/978-0-8176-4842-8_3 (https://doi.org/10.1007%2F978-0-8176-4842-8_3), ISBN 978-0-8176-4841-1. Exoo, G. (1989), "A lower bound for R(5,5)", Journal of Graph Theory, 13: 97–98, doi:10.1002/jgt.3190130113 (https://doi.org/10.1002%2Fjgt.3190130113). Graham, R.; Rothschild, B.; Spencer, J. H. (1990), Ramsey Theory, New York: John Wiley and Sons. Gross, Jonathan L. (2008), Combinatorial Methods with Computer Applications, CRC Press, p. 458, ISBN 978-1-58488-743-0 Harary, Frank (1972), Graph Theory, Addison-Wesley, pp. 16–17, ISBN 0-201-02787-9 Ramsey, F. P. (1930), "On a problem of formal logic", Proceedings of the London Mathematical Society, 30: 264–286, doi:10.1112/plms/s2-30.1.264 (https://doi.org/10.1112%2Fplms%2Fs2-3 15. Dushnik, Ben; Miller, E. W. (1941). "Partially ordered sets". American Journal of Mathematics. 63 (3): 600–610. doi:10.2307/2371374 (https://doi.org/10.2307%2F2371374). hdl:10338.dmlcz/100377 (https://hdl.handle.net/10338.dmlcz%2F100377). JSTOR 2371374 (htt ps://www.jstor.org/stable/2371374). MR 0004862 (https://www.ams.org/mathscinet-getitem?mr= 0004862).. See in particular Theorems 5.22 and 5.23. 16. Hirschfeldt, Denis R. (2014). Slicing the Truth. Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore. 28. World Scientific. 17. Diestel, Reinhard (2010). "Chapter 8, Infinite Graphs". Graph Theory (4 ed.). Heidelberg: Springer-Verlag. pp. 209–2010. ISBN 978-3-662-53621-6. 18. Smith, Warren D.; Exoo, Geoff, Partial Answer to Puzzle #27: A Ramsey-like quantity (http://Ra ngeVoting.org/PuzzRamsey.html), retrieved 2020-06-02 19. Wang, Hefeng (2016). "Determining Ramsey numbers on a quantum computer". Physical Review A. 93 (3): 032301. arXiv:1510.01884 (https://arxiv.org/abs/1510.01884). Bibcode:2016PhRvA..93c2301W (https://ui.adsabs.harvard.edu/abs/2016PhRvA..93c2301W). doi:10.1103/PhysRevA.93.032301 (https://doi.org/10.1103%2FPhysRevA.93.032301). References