1. Ramsey Numbers and Some Remarks on the
Theory of Graphs
Josh Mazen
April 27, 2015
Tulane University
Advisor: William Brian
Abstract
Ramsey theory presents many practical applications in the fields of combinatorics.
In this paper, we begin by reviewing some basic, key definitions of graph theory to
set up the main concepts. We then define the Ramsey number, an important
metric in graph theory, and explain the formal proof for the abridged Ramsey
Theorem as well as the upper and lower bounds for the Ramsey number of order
two. We apply Ramsey’s theorem to find a solution to the classic party problem:
what is the minimum number of people n at a party such that either x people
mutually know each other or y people mutually do not know each other? Further,
we extend Ramsey’s theorem to a specific three-coloring and find its Ramsey
number. Finally, we state two different variations of Ramsey’s unabridged theorem
and conclude with a remark on the nature of Ramsey numbers.
1 Introduction
Aside from his contributions to economics and philosophy, Frank Plumpton
Ramsey had a great interest in mathematical logic and combinatorics. In fact,
many of his constituents at Cambridge believed that he was only beginning to
make landmark contributions to his fields of interest before his death. He authored
1

2. ”On a Problem of Formal Logic” in 1930 with the intention of applying his
theorem, which he entitled Ramsey’s Theorem, to the field of mathematical logic.
He begins his paper with a proof of the infinite version of Ramsey’s Theorem, and,
as lengthy writing was considered to be more elegant during this era, his theorem
was quite extensive. However, his clarity and beauty still resonate with
combinatorial mathematicians to this day[2].
For the next seventeen years, Paul ˝Erdos and George Szekeres dissected
Ramsey’s famous theorem and the Ramsey Number in several papers that had
expanded upon his previous work. This lead to the beginning of an entirely new
branch of combinatorics known as Ramsey Theory, which has several applications
in the field of topology[1][2]. In this paper, we aim to simplify Ramsey’s Theorem
into a form that is easily imaginable, explain some of ˝Erdos’ work on discovering
the bounds for the Ramsey Number of order two, and apply combinatorial methods
to solve specific problems within these contexts.
2 Graph Theory Definitions and Notation
Before we delve into Ramsey’s Theorem, it is important to establish the basic
definitions of graph theory that will be relevant in understanding.
Definition 2.1: A graph G is defined as a finite, nonempty set V = V (G)
consisting of p vertices with a set X of q unordered pairs of distinct points of V [4].
With these points in some space, we would like to specify the relationship between
the vertices and this set X.
Definition 2.2 An edge, written as x = {u, v} ∈ X, is said to join two
separate vertices, u and v. It is also assumed, in the context of this paper, that
each pair of vertices will be connected by at most one edge. If two vertices are
connected by an edge, they are said to be adjacent[4].
2

3. Figure 2.3: a basic graph
With a general understanding of how graphs work, we can begin to describe specific
types of graphs that will be useful in solving specific problems related to Ramsey’s
Theorem. In a basic graph, vertices can either be adjacent or not adjacent. A
special case of a basic graph is one such that all vertices are adjacent.
Definition 2.4 A complete graph of n points, written as Kn, is a graph in
which every pair of vertex is connected by an edge[4].
K4 K5 K6
Figure 2.5: Three complete graphs
Now suppose, for example, we would like to examine only the outer edges of K5 as
shown in Figure 2.5. We will use the following definition:
Definition 2.6: A subgraph of G, intuitively, is a graph that has all of its
vertices and edges contained within G[4].
3

4. Figure 2.7: one subgraph of K5
It is important to note that there are many subgraphs of any single graph, which
we will delve into more deeply later in this paper. Suppose we have two graphs with
the same number of points. If they appear to have similar structure, does this mean
that they are the same or does the appearance of the graph make them different?
Similar to the idea of congruency in geometry, there is a definition in graph theory:
Definition 2.8: A graph G is said to be isomorphic with another graph H if
there exists a one-to-one mapping of each of its vertices. In other words, the
adjacencies that exist between the vertices in G correspond with those in H[4].
Figure 2.9: Two graphs isomorphic to each other
In our previous examples, several of the graphs had many pairs of vertices that
were not joined by an edge, with some vertices not having any adjacencies at all.
To address these relations that appear not to exist, we will make another definition:
Definition 2.10: The complement of a graph G, written as ¯G, is the graph
such that if a pair of points are connected by an edge in G, they are not connected
in ¯G. Furthermore, if the graphs G and ¯G are isomorphic with each other, then
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5. both graphs are said to be self-complementary[4].
Figure 2.11: The smallest self-complementary graph
Adjacency can be interpreted as two vertices being connected by an edge. However,
there are other ways to graphically represent that two points do, in fact, share a
relationship. For example, it would suffice if we were to connect every vertex on a
graph and let the color red represent an adjacency between two points and for blue
to represent the complement of red. Furthermore, we can represent different types
of relationships with multiple colors, such as to describe being strongly related,
somewhat related, weakly related, or not related. We define a coloring as follows:
Definition 2.12: Suppose we have a graph G. An r-coloring of the set V
maps as such:
χ : V → [r]
for some v ∈ V , χ(v) is the color of v[2].
Now that we have clearly defined all the necessary terms, we can begin to
uncover Ramsey’s Theorem and interpret its meaning as it relates to graph theory.
3 Ramsey’s Abridged Theorem
Before we introduce Ramsey’s Theorem, it is important to familiarize with the
notation used. Let [n] denote the set of natural numbers up to n. Then we
establish to following property:
Property 3.1: n → (l) denotes that, given any 2-coloring of Kn, a set
T ⊆ [n], |T| = l such that every edge in T is monochromatic. Further generalizing
this notation, n → (l1, l2, ...lr) denotes that, for every r-coloring of Kn, a set
5

6. T ⊆ [n], |T| = li exists such that every edge in T is colored i, for 1 ≤ i ≤ r. If
l1 = l2 = ... = lr, then we abbreviate with n → (l)r[2][4]. We will note some
important trivialities about this notation:
1. If li ≤ li for 1 ≤ i ≤ r, and n → (l1, l2, ...lr), then n → (l1, l2, ...lr).
2. If m ≥ n and n → (l1, l2, ...lr), then m → (l1, l2, ...lr).
3. Let σ be a permutation of [r]. Then n → (l1, l2, ...lr) iff n → (lσ1, lσ2, ...lσr)
4. n → (l1, l2, ...lr) iff n → (l1, l2, ...lr, 2). In particular, l1 → (l1, 2).
Example 3.2: Let our first color be red and the second blue. Then
10 → (4, 3) asserts that, given any ten vertices on a graph, either four of them are
mutually connected by red edges or three are mutually connected by blue edges. It
would follow from triviality 1 that 10 → (4, 3) implies 10 → (3, 3), since we could
simply ignore one red edge and the statement would still be true. Similarly,
10 → (4, 3) implies 11 → (4, 3) by triviality 2 if we are to simply ignore the 11th
vertex. If 10 → (4, 3), then 10 → (3, 4) if we simply interchange the coloring.
Lastly, 10 → (4, 3) would imply that 10 → (4, 3, 2), for we could always connect two
vertices by a yellow edge as well (as long as there are more than two vertices)[2].
Knowing these trivialities, it would be useful to have a minimum n such that
the notation described above were true so that we might be able to understand
these numbers more effectively. This is precisely what we are searching for when we
speak of the Ramsey Number.
Definition 3.3: A Ramsey Number, R(l1, ..., lr) is defined as the smallest n
such that
n → (l1, l2, ...lr).
Following directly from our definition, we have our first theorem[2].
Theorem 3.4 (Ramsey’s Theorem Abridged): The Ramsey Number R
is well defined for all l1, ..., lr ∈ N. In other words, for all l1, ..., lr, there exists n
such that
n → (l1, l2, ...lr).
In this particular case, we will give the proof for r = 2.
Proof: It is trivial to say that the Ramsey Number exists if either l1 = 1 or
l2 = 1, for any single point will always be monochromatic. By this logic, the
6

7. Ramsey number for the case where either l1 = 1 or l2 = 1 is equal to 1.
Additionally, as we have shown by triviality 4 from Definition 3.1, we know that the
Ramsey Number is well defined if either l1 = 2 or l2 = 2. It is therefore necessary to
show that the Ramsey Number is defined for either l1, l2 ≥ 3. Now we will assume,
by using a double induction, that R(l1, l2 − 1) and R(l1 − 1, l2) both exist.
We will prove the following recursion to show that the Ramsey Number is well
defined:
n = R(l1, l2 − 1) + R(l1 − 1, l2) → (l1, l2)
Firstly, we will fix a 2-coloring red and blue χ of Kn, and we will also fix an edge
by the inductive step. To clarify the goal of our hypothesis, we must be reminded
that the Ramsey Number R(l1, l2) is defined as the smallest natural number such
that if Kn is 2-colored, then the following properties are satisfied:
1. There is a red copy of Kl1 , or
2. There is a blue copy of Kl2 .
If the first edge is set as red, then the remaining possible colorations can be
written as R(l1 − 1, l2), since this is the smallest natural number that would allow
for the subgraphs described above, excluding the edge that has been fixed red.
Symmetrically, if the first edge is blue, then the remaining colorations are written
as R(l1, l2 − 1). Since only one of these expressions can be true in a given graph,
the total number of ways in which this can be done can be expressed by adding the
expressions, i.e. n = R(l1, l2 − 1) + R(l1 − 1, l2). Therefore, this expression maps to
the Ramsey Number n = R(l1, l2)[2].
We have successfully shown that the Ramsey Number is always well defined,
but this begs the question: what are the properties of the Ramsey Number? We
will show some of its properties in the following section that will be derived from
the theorem we have just proven.
4 Bounds of Ramsey Numbers
While our theorem does not give us an exact value for each Ramsey Number, it
directly gives rise to an important property that will help us to determine the
upper bound for the Ramsey number of order 2:
R(l1, l2) ≤ R(l1, l2 − 1) + R(l1 − 1, l2)[2]
7

8. The upper bounds for the Ramsey Numbers up to R(6, 6) following this recursion
are written in the table below:
Table 4.1
l1l2 1 2 3 4 5 6
1 1 1 1 1 1 1
2 1 2 3 4 5 6
3 1 3 6 10 15 21
4 1 4 10 20 35 56
5 1 5 15 35 70 126
6 1 6 21 56 126 252
It would help to find a closed-form formula for the Ramsey number of degree two.
If we examine Table 4.1, we can see that the diagonals of the table form Pascal’s
famous triangle for the binomial coefficient[6]. The identity for the non-recursive
upper bound is as follows:
Identity 4.2: R(l1, l2) ≤ l1+l2−2
l1−1
Proof: The closed-form formula can be derived using double induction. For
the base case, let l1 = l2 = 2. Clearly R(2, 2) = 2 ≤ 2+2−2
2−1
= 2. Assuming the
expression holds for R(l1 − 1, l2) and R(l1, l2 − 1), then from our recursive formula,
we say R(l1, l2) ≤ R(l1, l2 − 1) + R(l1 − 1, l2) ≤ l1+l2−3
l1−2
+ l1+l2−3
l1−1
= l1+l2−2
l1−1
[5][6].
Naturally, if we have an upper bound, it would also be helpful to have a lower
bound for the Ramsey number. While the lower bound for R(l1, l2) does not have
an apparent closed formula for l1 = l2, it does for l1 = l2 = l.
Theorem 4.3 Let l ≥ 3. Then
2l/2
< R(l, l) ≤
2l − 2
l − 1
< 4l−1
.[1]
Proof: The second inequality follows directly from Identity 4.2 and the third
inequality can be proven very easily by induction, so we focus on the proof for the
8

9. first inequality. If our graph has n vertices, then n ≤ 2l/2
, which we will have shown
by the end of the proof. Because the vertices of the graphs are distinguishable, the
number of different graphs of n vertices that can be constructed are 2(n
2) because
we can either choose to connect two vertices by an edge of one color or the other
and select two vertices from n to connect from that given color.
To find the number of different graphs of n which contain a complete
monochromatic graph of order l, we will take the total number of graphs and
remove all subgraphs of size l, which equates to 2(n
2)/2(l
2). In other words, we
choose any l points, then fill the rest of the graph with 2(n
2)−(l
2). Thus, the number
of graphs with n ≤ 2l/2
vertices containing a complete graph of order l is, by simple
algebraic manipulation, less than
n
l
2(n
2)−(l
2) <
nl
l!
2(n
2)−(l
2) <
2(n
2)
2
By a simple inductive calculation for l ≥ 3, 2nl
< l!2(l
2), . It follows from this
inequality that there exists a graph such that neither it nor its complementary
graph contains a complete subgraph of order l, and clearly fewer than half of the
potential graphs contain either a complete subgraph or an empty subgraph.
Therefore, the bound holds[1].
The lower bounds for the Ramsey numbers we have just described are as
follows, rounded up to the nearest integer:
Table 4.4
l1l2 1 2 3 4 5 6
1 N/A
2 N/A
3 3
4 4
5 6
6 8
Directly from this theorem, we arrive at another important inequality that details
then nature of our order, l in relation to our vertices, n.
9

10. Property 4.5: Define A(n) as the greatest integer such that given graph G
with n vertices, either it or its complementary graph contains a complete subgraph
of order A(n). In other words, if R(l, l) = n ⇔ A(n) = l. Then for A(n) ≥ 3:
log n
2 log 2
< A(n) <
2 log n
log 2
[1]
Proof: By Theorem 4.3, 2A(n)/2
< n < 4A(n)−1
< 4A(n)
. Taking the log of
both sides, we obtain
A(n)
2
< log n < A(n) log 4.
With some quick arithmetic, the solution to this theorem becomes evident[1].
With these bounds for the Ramsey Numbers, we have somewhat of an
estimate for their values. However, it would naturally be much better to have an
exact Ramsey Number. In the following section, we will illustrate specific examples
of Ramsey Numbers and how they might be derived combinatorially.
5 The Six Person Problem and a Three-Colored
Example
One of the most practical applications for the use of Ramsey Numbers is the party
problem. The problem is as follows: If there are l1 people at a party that mutually
know each other and l2 that mutually do not know each other, what is the minimum
number of guests at the party? One such example has been most famously proven
time and again, as it is one of the simplest to explain to the average individual
unfamiliar with ideas in Ramsey Theory.
Theorem 5.1: If there are six guests at a party, then either three of them
mutually know each other or three of them mutually do not know each
other[3][4][6].
Proof:The upper bound for R(3, 3) is 6, as we can see from Table 4.1. In
order to show that R(3, 3) = 6, we must show that we can create a 2-coloring of K5
that does not contain a monochromatic K3, i.e. R(3, 3) > 5, for this would be a
direct contradiction of the definition of Ramsey Number. If we can draw a graph
that contains 5 vertices and does not contain a monochromatic triangle, then we
have completed our goal. Let us represent two people ”knowing each other” by two
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11. vertices being connected by a red line, and the opposite by a blue line. The most
famous drawing of this solution is as such:
Figure 5.2: A complete K5 that does not contain a monochromatic
K3[3][6]
As we can see from Figure 5.2, from a five point graph, we cannot create a
monochromatic K3 from five vertices, since no single monochromatic triangle exists.
Furthermore, since the graph is self-complementary, this will hold for any case of
five vertices. Therefore, Theorem 5.1 is true and R(3, 3) = 6.
Similar processes have been done to find the exact Ramsey numbers for
several other pairs of natural numbers. The complete list of known Ramsey
Numbers (without triviality for the cases where either l1 or l2 = 1 or 2) is listed
below:
Table 5.3
l1l2 1 2 3 4 5 6 7 8 9
1 1 1 1 1 1 1 1 1 1
2 1 2 3 4 5 6 7 8 9
3 1 3 6 9 14 18 23 28 36
4 1 4 9 18 25
5 1 5 14 25
6 1 6 18
7 1 7 23
8 1 8 28
9 1 9 36
11

12. Now that we have discussed particular two-colorings Ramsey Numbers, naturally
we ask about those of higher colorings. The Ramsey Numbers of higher coloring
have been studied less so than those of two colors, partially because we do not
know much of these numbers and can rarely use a strong bound. Some bounds
have been approximated for specific Ramsey Numbers, but there aren’t many
concrete examples which dictate strong bounds or which have a definitive solution.
However, one example has been famously shown using Theorem 5.1:
Theorem 5.4: R(3, 3, 3) = 17.
Proof: We would like to find a graph such that there are no monochromatic
triangles to find the upper bound. Suppose we use the colors red, green, and
yellow. We will first fix a vertex v. If we consider all of the vertices which connect
to v with a green edge, we will call them the green neighborhood. The green
neighborhood cannot contain any further green edges, since there would otherwise
be a green triangle consisting of the endpoints of that green edge and v. Therefore,
any edge connected to the green neighborhood must be either red or yellow.
If we want to be sure that the green neighborhood does not contain any
triangles, we know that there can only be at most 5 vertices in the green
neighborhood since R(3, 3) = 6 by Theorem 5.1 and we either choose to color the
connecting edges green or not green. Without loss of generality, the red and yellow
neighborhood of be can also only have at most 5 vertices. Since every vertex is
either in the red, yellow, or green neighborhood of v, with the exception of v itself,
the entire complete graph can only have at most 1 + 5 + 5 + 5 = 16 vertices.
Therefore, including v, R(3, 3, 3) ≤ 17[3].
If we draw K16, we see that it is possible to draw two 3-colorings that do not
contain a monochromatic triangle.
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13. Figure 5.5: The only two known drawings that show K16 without any
monochromatic K3 for three different colorings[3]
6 Ramsey’s Unabridged Theorem and
Conclusion
We have given an overview of the Ramsey Number as it exists in two dimensions,
but this is only the tip of the iceberg when it comes to this perplexing phenomena.
Graphs can exist within three dimensions, where an ”adjacency” can be defined as
three vertices in space being connected by a triangle. Further, they can exist within
four dimensions, where adjacency is connection by a tetrahedron, and in even more
complex shapes beyond the fourth dimension[2]. Ramsey’s famous theorem, in
13

14. which he proved the unabridged version for all colorings and all dimensions, has
been interpreted in so many ways that numerous mathematicians have created
multiple definitions for the theorem. We will state a few different interpretations of
Ramsey’s full Theorem to introduce the work on Ramsey Theory that is still being
studied today. First, following our notation in Section 2, we will make a few
modifications to generalize Ramsey’s Theorem.
Property 6.1: n → (l1, l2, ...lr)k
denotes that, for every r-coloring of [n]k
,
where [n]k
= {Y : Y ⊂ {1, ...n}, |Y | = k}, a set T ⊆ [n], |T| = li exists such that
[T]k
is colored i, for 1 ≤ i ≤ r[2].
This establishes our arrow notation as it exists in infinitely many dimensions.
Ramsey’s Theorem, in this regard, is defined as follows:
Theorem 6.2 The Ramsey Number is well defined such that for all k, l1, ...lr,
there exists an n0 ≤ n so that
n → (l1, l2, ...lr)k
[2].
While we will not discuss the formal proof for the unabridged theorem in this
paper, it is proven by using induction and the Pigeon-Hole Principle[2].
Additionally, Paul ˝Erdos had his own interpretation of Ramsey’s Theorem;
however, his version of the theorem might appear to be referring to a completely
different concept to the new reader. ˝Erdos’ Unabridged Ramsey’s Theorem is
stated as follows:
Theorem 6.3: Suppose there exists a graph with n vertices, and let
i, k, l ∈ N, i ≤ k, i ≤ l. There exists a function f(i, k, l) such that, if n ≥ f(i, k, l)
and there is a given collection of combinations of order i of these n vertices, such
that every combination of order k contains at least one combination of order i in
the collection of n vertices, then there exists a combination of order l such that all
combinations of l of order i belong to the collection[1].
Again, we will not prove the theorem in this paper; however, similarly to our
proof for the abridged Ramsey Theorem, the unabridged version is done through a
multi-inductive process. It is proven by showing that
f(i, k, l) ≤ f(i − 1, f(i, k − 1, l), f(i, k, l − 1)) + 1.[1]
14

15. As it has been made evident in this paper, there is still much work that needs
to be done to uncover the true nature of Ramsey Numbers. Very little, if anything,
is known about Ramsey Numbers of higher order, and there does not appear to be
a clear path in sight. Since the conception of the Ramsey Number, many have
searched for proofs of specific Ramsey numbers, as it is widely believed that a
closed formula does not exist. In particular, most combinatorial mathematicians
have searched for proofs for 2-colorings of two-dimensional Ramsey numbers, as
they are the simplest to understand, yet the solution is not obvious. Joel Spencer,
a famous combinatorial mathematician, humorously describes the nature of this
mathematical phenomenon, in saying, ”...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 this case...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...we should attempt to destroy the aliens.”
References
[1] P. ˝Erdos: Some Remarks on the Theory of Graphs, Bulletin of the American
Mathematical Society, 53, 292-294 (1947).
[2] R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, Second
Edition, Wiley-Interscience Publications, 1990.
[3] R. E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic
Graphs, Canadian Journal of Mathematics, 7, 1-7 (1955).
[4] F. Harary, Graph Theory, Perseus Books, 1969.
[5] D. Samana and V. Longani: Upper Bounds of Ramsey Numbers, Applied
Mathematical Sciences, 6:98, 4857-4861 (2012).
[6] G. Szekeres and P. ˝Erdos, A combinatorial problem in geometry, Compositio
Math, 2, 463-470 (1935).
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