User:Zscoder/sandbox

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 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 $K_{L−1}$ with no blue $K_{r}$ subgraph and no red $K_{s}$ subgraph. Such a counterexample is called a Ramsey graph. Brendan McKay maintains a list of known Ramsey graphs. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence.

Computational complexity
"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."

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 $K_{n}$ has $1⁄2n(n − 1)$ edges, so there would be a total of $c^$ graphs to search through (for $c$ colours) if brute force is used. Therefore, the complexity for searching all possible graphs (via brute force) is $O(c^{n^{2}})|undefined$ for $c$ colourings and at most $n$ nodes.

The situation is unlikely to improve with the advent of quantum computers. The best known algorithm 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.

Known values
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 $s$-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, 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. (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.

The exact value of $R(5, 5)$ is unknown, although it is known to lie between $43$ (Geoffrey Exoo (1989) ) and $R(5, 5)$ (Angeltveit and McKay (2017) ) (inclusive).

In 1997, McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture that $48$. They were able to construct exactly 656 $R(5, 5) = 43$ graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a $(5, 5, 42)$ graph.

For $(5, 5, 43)$ with $R(r, s)$, only weak bounds are available. Lower bounds for $r, s > 5$ and $R(6, 6)$ have not been improved since 1965 and 1972, respectively.

$R(8, 8)$ with $R(r, s)$ are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. $r, s ≤ 10$ with $R(r, s)$ are given by $r, s < 3$ and $R(1, s) = 1$ for all values of $R(2, s) = s$.

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. 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 $s$.)

Asymptotics
The inequality $R(r, s) = R(s, r)$ 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 $R(r, s)$,
 * $$R(s, s) \leq [1 + o(1)]\frac{4^{s - 1}}{\sqrt{\pi s}}.$$

An exponential lower bound,
 * $$R(s, s) \geq [1 + o(1)] \frac{s}{\sqrt{2} e} 2^{s/2},$$

was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is obviously a huge gap between these two bounds: for example, for $R(r, s) ≤ R(r − 1, s) + R(r, s − 1)$, this gives $r = s$. Nevertheless, exponential growth factors of either bound have not been improved to date and still stand at $s = 10$ and $101 ≤ R(10, 10) ≤ 48620$ respectively. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers currently stand at
 * $$[1 + o(1)] \frac{\sqrt{2} s}{e} 2^{\frac{s}{2}} \leq R(s, s) \leq s^{-(c \log s)/(\log \log s)} 4^s,$$

due to Spencer and Conlon respectively.

For the off-diagonal Ramsey numbers $4$, it is known that they are of order $$\tfrac{t^2}{\log t}$$; this may be stated equivalently as saying that the smallest possible independence number in an $n$-vertex triangle-free graph is
 * $$\Theta \left (\sqrt{n\log n} \right ).$$

The upper bound for $√2$ 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, $R(3, t)$, with $s$ fixed and $t$ growing, the best known bounds are
 * $$c'_s \frac{t^{\frac{s + 1}{2}}}{(\log t)^{\frac{s + 1}{2} - \frac{1}{s - 2}}} \leq R(s, t) \leq c_s \frac{t^{s - 1}}{(\log t)^{s - 2}},$$

due to Bohman and Keevash and Ajtai, Komlós and Szemerédi respectively.

Induced Ramsey
There is a less well-known yet interesting analogue of Ramsey 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 Erdös, Hajnal and Pósa, Deuber and Rödl in the 1970's.  Since then, there has been much research in obtaining good bounds for induced Ramsey numbers.

Statement
Let $$H$$ be a graph on $$n$$ vertices. Then, there exists a graph $$G$$ such that any coloring of the edges of $$G$$ in two colors contains a monochromatic induced copy of $$H$$ (i.e. an induced subgraph of $$G$$ such that it is isomorphic to $$H$$ and its edges are monochromatic). The smallest possible number of vertices of $$G$$ is the induced Ramsey number $$r_{\text{ind}}(H)$$.

Sometimes, we also consider the asymmetric version of the problem. We define $$r_{\text{ind}}(X,Y)$$ to be the smallest possible number of vertices of a graph $$G$$ such that every coloring of the edges of $$G$$ in red or blue contains a red induced subgraph of $$X$$ or blue induced subgraph of $$Y$$.

History and Bounds
Similar to Ramsey's Theorem, it is unclear a priori whether induced Ramsey numbers exist for every graph $$H$$. In the early 1970's, Erdös, Hajnal and Pósa, Deuber and Rödl independently proved that this is the case. However, the original proofs gave terrible bounds on the induced Ramsey numbers. It is interesting to ask if better bounds can be achieved. In 1974, Erdös conjectured that there exists a constant $$c$$ such that every graph $$H$$ on $$k$$ vertices satisfies $$r_{\text{ind}}(H) \le 2^{ck}$$. If this conjecture is true, it would be optimal up to the constant $$c$$ 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, Erdös and Hajnal claimed that they proved the bound $$r_{\text{ind}}(H) \le 2^{2^{k^{1+o(1)}}}$$. However, that was still far from the exponential bound conjectured by Erdös. It was not until 1998 when a major breakthrough was achieved by Kohayakawa, Prőmel and Rődl, who proved the first almost-exponential bound of $$r_{\text{ind}}(H) \le 2^{ck(\log k)^2}$$ for some constant $$c$$. Their approach was to consider a suitable random graph constructed on projective planes and show that it has the desired properties with nonzero probability. The idea of using random graphs on projective planes have also previously been used in studying Ramsey properties with respect to vertex colorings and the induced Ramsey problem on bounded degree graphs $$H$$.

Kohayakawa, Prőmel and Rődl'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. In fact, they showed that every $(n,d,\lambda)$-graph $$G$$ with small $$\lambda$$ and suitable $$d$$ contains an induced monochromatic copy of any graph on $$k$$ vertices in any coloring of edges of $$G$$ in two colors. In particular, for some constant $$c$$, the Paley graph on $$n \ge 2^{ck\log^2 k}$$ vertices is such that all of its edge colorings in two colors contain an induced monochromatic copy of every $$k$$-vertex graph.

In 2010, Conlon, Fox and Sudakov were able to improve the bound to $$r_{\text{ind}}(H) \le 2^{ck\log k}$$, which remains the current best upper bound for general induced Ramsey numbers. Similar to the previous work in 2008, they showed that every $(n,d,\lambda)$-graph $$G$$ with small $$\lambda$$ and edge density $$\frac{1}{2}$$ contains an induced monochromatic copy of every graph on $$k$$ vertices in any edge coloring in two colors. Currently, Erdös's conjecture that $$r_{\text{ind}}(H) \le 2^{ck}$$ remains open and 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).

Proof Sketch
There are many proofs of the induced Ramsey theorem, though given the nature of the problem they are more difficult than the original Ramsey's Theorem. Elementary proofs exist, but they are quite complicated and involved (for example, see ). Instead, we will give an outline of a more standard proof using Szemerédi regularity lemma (presented in ).

Let $$V(G),E(G)$$ denote the vertex and edge set of the graph $$G$$. For a pair of vertex sets $$X, Y \in V(G)$$, define $$e(X,Y) = \{(x,y): (x,y) \in E(G)\}$$ and $$d(X,Y) = \frac{e(X,Y)}{|X||Y|}$$ (edge density between $$X$$ and $$Y$$).

Fix a graph $$H$$ on $$h$$ vertices. We show that an Erdos–Renyi random graph with $$N$$ vertices and edge density $$p=\frac{1}{2}$$ works with high probability for large enough $$N$$.

Choose a Erdos–Renyi random graph $$G(N,\frac{1}{2})$$ (where $$N$$ is large). Let $$\epsilon>0$$ be a small constant we will choose later. By Chernoff inequality and union bound, we can prove that with high probability, all constant-fraction sized vertex subsets $$A,B$$ satisfy $$|d(A,B)-\frac{1}{2}| \le \frac{\epsilon}{2}$$.

Now, consider a choice of $$G$$ such that the above condition holds and an arbitrary coloring of the edges of $$G$$ in red and blue. We apply the equitable version of Szemerédi regularity lemma to the subgraph of $$G$$ consisting of the blue edges. Let $$V_1,V_2,\cdots,V_M$$ be a $$\frac{\epsilon}{2}$$-regular equipartition generated by the regularity lemma (in particular, sizes of all sets differ by at most $$1$$). Since most pairs of vertex sets are $$\frac{\epsilon}{2}$$-regular, if $$\epsilon$$ is small enough and $$M$$ is large enough, by Turan's Theorem we can find $$m$$ sets (suppose without loss of generality they are $$V_1,V_2,\cdots,V_m$$) such that every pair is $$\frac{\epsilon}{2}$$-regular in the blue edges. A key observation is that for our graph $$G$$, if $$(A,B)$$ is $$\frac{\epsilon}{2}$$-regular in the blue edges, it is $$\epsilon$$-regular in the blue edges (we use the fact that the density of each pair of vertex subsets is close to $$\frac{1}{2}$$). Hence, $$(V_i,V_j)$$ is $$\epsilon$$-regular in both colors for all $$1 \le i < j \le m$$.

Set $$\epsilon \le 0.02$$. Then, $$d(V_i,V_j) \in [0.49,0.51]$$ for all $$1 \le i < j \le m$$. Consider a complete graph $$R$$ on $$m$$ vertices and color an edge $$(i,j)$$ yellow if the density of blue edges between $$V_i,V_j$$ is at least $$0.25$$ and green otherwise. By Ramsey's Theorem, as long as $$m \ge 4^h$$, there exists $$h$$ subsets (without loss of generality, $$V_1,V_2,\cdots,V_h$$) such that every pair of them has the same edge color in $$R$$. Without loss of generality, suppose it is yellow.

To finish, we need to construct the induced subgraph $$H$$ from a bunch of large enough pairwise regular sets with non-negligible density.

Lemma 3. Let $$H$$ be a graph on $$h$$ vertices. There exists some constants $$s, \epsilon$$ depending on $$h$$ such that if $$V_1,V_2,\cdots,V_h$$ are disjoint vertex sets of a graph $$G$$ with size $$\ge s$$, $$(V_i,V_j)$$ is $$\epsilon$$-regular and $$d(V_i,V_j) \in \left[\frac{1}{8},\frac{7}{8}\right]$$ for every $$1 \le i < j \le h$$, then $$G$$ contains an induced copy of $$H$$ with exactly one vertex in each $$V_i$$.

Proof Sketch. The idea is to embed the vertices of $$H$$ one by one and maintain the set of possible vertices you can use for the embedding. Initially, let $$S_1,S_2,\cdots,S_h$$ be such that $$S_i = V_i$$. Suppose we embed the first $$i-1$$ vertices of $$H$$ into $$V_1,V_2,\cdots,V_{i-1}$$. We will embed the $$i$$-th vertex into $$S_{i}$$. For every $$j>i$$, we look at the current $$S_i$$ and replace it with $$S_{i}' \subset S_i$$ such that the size of $$S_i'$$ is at least a constant fraction of the original set $$S_i$$ and each vertex of $$S_i$$ has at least $$\frac{1}{16}|S_{j}|$$ neighbors in $$S_j$$ if $$(i,j) \in E(H)$$ and at least $$\frac{1}{16}|S_{j}|$$ non-neighbors in $$S_j$$ if $$(i,j) \not\in E(H)$$. This is possible when $$\epsilon$$ is small enough due to regularity and the density condition between the pair. After repeating the process for all $$i$$, we choose a vertex $$v_i$$ from $$S_i$$ to embed the $$i$$-vertex of $$H$$, and update all $$S_j$$s with $$j>i$$ accordingly (i.e. if $$(i,j) \in E(H)$$, replace $$S_j$$ with the subset of all vertices neighboring $$v_i$$ and the complement otherwise). For $$s$$ large enough and $$\epsilon$$ small enough (with respect to $$h$$), this process will work. This completes the description of the embedding and the proof of the lemma.

To finish the proof of the original Induced Ramsey's Theorem, we consider only blue edges between $$V_i,V_j$$ with $$(i,j) \in E(H)$$ and both blue and red edges between $$V_i,V_j$$ with $$(i,j) \not\in E(H)$$ and apply Lemma 3. This allows us to find an induced monochromatic copy of $$H$$ and completes the proof.

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). In particular, many of these classes have induced Ramsey numbers polynomial in the number of vertices.

If $$H$$ is a cycle, path or star on $$k$$ vertices, it is known that $$r_{\text{ind}}(H)$$ is linear in $$k$$.

If $$H$$ is a tree on $$k$$ vertices, it is known that $$r_{\text{ind}}(H) = O(k^2\log^{2}k)$$. It is also known that $$r_{\text{ind}}(H)$$ is superlinear (i.e. $$r_{\text{ind}}(H) = \omega(k)$$). Note that this is in contrast to the usual Ramsey numbers, where the Burr–Erdős conjecture (now proven) tells us that $$r(H)$$ is linear (since trees are 1-degenerate).

For graphs $$H$$ with number of vertices $$k$$ and bounded degree $$\Delta$$, it was conjectured that $$r_{\text{ind}}(H) \le cn^{d(\Delta)}$$, for some constant $$d$$ depending only on $$\Delta$$. This result was first proven by Luczak and Rődl in 1996, with $$d(\Delta)$$ growing as a tower of twos with height $$O(\Delta^2)$$. More reasonable bounds for $$d(\Delta)$$ were obtained since then. In 2013, Conlon, Fox and Zhao showed using a counting lemma for sparse pseudorandom graphs that $$r_{\text{ind}}(H) \le cn^{2\Delta + 8}$$, where the exponent is best possible up to constant factors.

Generalizations
Similar to Ramsey numbers, we can generalize the notion of induced Ramsey numbers to hypergraphs and multicolor settings.

More colors
We can also generalize induced Ramsey theorem to a multicolor setting. For graphs $$H_1,H_2,\cdots,H_r$$, define $$r_{\text{ind}}(H_1,H_2,\cdots,H_r)$$ to be the minimum number of vertices in a graph $$G$$ such that any coloring of the edges of $$G$$ into $$r$$ colors contain a induced subgraph isomorphic to $$H_i$$ where all edges are colored in the $$i$$-th color for some $$1 \le i \le r$$. Let $$r_{\text{ind}}(H;q) := r_{\text{ind}}(H,H,\cdots,H)$$ ($$q$$ copies of $$H$$).

It is possible to derive a bound on $$r_{\text{ind}}(H;q)$$ which is approximately a tower of two of height $$\sim \log q$$ by iteratively applying the bound on the two-color case. The current best known bound is due to Fox and Sudakov, which achieves $$r_{\text{ind}}(H;q) \le 2^{ck^3}$$, where $$k$$ is the number of vertices of $$H$$ and $$c$$ is a constant depending only on $$q$$.

Hypergraphs
We can extend the definition of induced Ramsey numbers to $$d$$-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 $$H$$ be a $$d$$-uniform hypergraph with $$k$$ vertices. Define the tower function $$t_r(x)$$ by letting $$t_1(x) = x$$ and for $$i \ge 1$$, $$t_{i+1}(x) = 2^{t_{i}(x)}$$. Using the hypergraph container method, Conlon, Dellamonica, La Fleur, Rödl and Schacht were able to show that for $$d \ge 3, q \ge 2$$, $$r_{\text{ind}}(H;q) \le t_d(ck)$$ for some constant $$c$$ depending on only $$d$$ and $$q$$. In particular, this result mirrors the best known bound for the usual Ramsey number when $$d=3$$.

Infinite graphs
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.


 * 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.

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 $$[k]^{(n)}$$ without a monochromatic set of size T. Let Ck denote the c-colourings of $$[k]^{(n)}$$ without a monochromatic set of size T.

For any k, the restriction of a colouring in Ck+1 to $$[k]^{(n)}$$ (by ignoring the colour of all sets containing k + 1) is a colouring in Ck. Define $$C^1_k$$ to be the colourings in Ck which are restrictions of colourings in Ck+1. Since Ck+1 is not empty, neither is $$C^{1}_k$$.

Similarly, the restriction of any colouring in $$C^1_{k+1}$$ is in $$C^1_k$$, allowing one to define $$C^2_k$$ as the set of all such restrictions, a non-empty set. Continuing so, define $$C^{m}_k$$ for all integers m, k.

Now, for any integer k, $$C_k\supseteq C^1_k\supseteq C^2_k\supseteq \cdots$$, and each set is non-empty. Furthermore, Ck is finite as $$|C_k|\le c^{\frac{k!}{n!(k-n)!}}$$. It follows that the intersection of all of these sets is non-empty, and let $$D_k=C_k\cap C^1_k\cap C^2_k\cap \cdots$$. 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 $$\mathbb N^{(n)}$$ 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.

Hypergraphs
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.

Directed graphs
It is also possible to define Ramsey numbers for directed graphs; these were introduced by. 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 34 ≤ R(7) ≤ 47.

Relationship to the axiom of choice
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. Over ZF, however, the graph version is equivalent to the classical Kőnig's lemma.