RAMSEY THEORY FOR GRAPHS
FABRICIO SIQUEIRA BENEVIDES
A BSTRACT. “In any party of six people there is always three people who are all
either mutual acquaintances or mutual strangers”. This saying is the simplest consequence of Ramsey’s Theorem. Ramsey Theory is a branch of Extremal Combinatorics. Its problems have been much studied over the last decades and appear in
various forms, with earlier results dating back to 1928. We focus on recent results
for graphs, assuming no previous knowledge of Ramsey Theory. We shall discuss
some of the common techniques applied in those results. For example, the acclaimed
Szemerédi’s Regularity Lemma.
C ONTENTS
1. Introduction
1.1. Notation
1.2. Graphs
1.3. Ramsey Theorems in Graph Theory
1.4. Ramsey Theory outside Graph Theory
1.5. Bounds on Ramsey numbers
1.6. Density Theorems (Turán/Szemerédi)
Exercises
2. Szemerédi’s Regularity Lemma
2.1. The Regularity Lemma for Graphs
2.2. Embeddings Lemmas
Exercises
3. Applying the regularity lemma
3.1. Triangle removal lemma
3.2. A proof of Roth’s theorem
3.3. Two standard applications in Graph Theory
Exercises
4. Ramsey numbers of cycles
4.1. From connected matchings to paths
5. Futher reading and acknowledgements
References
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1. I NTRODUCTION
Very generally, one can say that Ramsey Theory is a branch of mathematics that
study conditions for which certain “order” must appear. Some like to say that Ramsey Theory tells that “there is no absolute disorder”. In these notes, we will first
Date: 30 of may 2013.
1991 Mathematics Subject Classification. 05D10.
Acknowledgments.
1
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FABRICIO SIQUEIRA BENEVIDES
introduce Ramsey Theory with a series of examples showing what we mean by “order” in various set-ups. But we will keep our focus on Ramsey theory for graphs.
We will also study a few related problems in Extremal Combinatorics, like Turántype problems and the problem of finding long arithmetic progressions in sets with
positive density. We will study some important tools, like the Regularity lemma and
the Triangle removal lemma, which are extremely useful not only in Ramsey Theory,
but also e various branches of combinatorics. And we will see a few applications of
those tools.
In 1928, the young mathematician Frank Plumpton Ramsey [18] wrote an article
about an algorithm problem in propositional logic. In that article, Ramsey proved
also a purely mathematical result, well-known nowadays as the Ramsey’s theorem.
The simplest particular case of this theorem yields the following well known curious
fact: in a party of six people there is always 3 of them that know each other or 3 of
them that do not know each other. Ramsey’s theorem was at first only a tool in the
original article but have turned out to be more acknowledged than the article itself
and has given risen for an whole new area. We remark that some early theorems in
Ramsey theory actually date before the result of Ramsey.
The original theorem of Ramsey has been expanded and applied to a number
of areas in Mathematics including areas outside Combinatorics. It involves a wide
number of techniques which are now part of what is known as Ramsey Theory.
Notably in the past three decades, Ramsey theory has evolved from a collection of
theorems to become a cohesive sub-area of Extremal Combinatorics. One can find
full books on the topic, for example, the one by Graham [9]. Nevertheless, a number
of the original problems are still unsolved.
1.1. Notation. Our notation is standard (e.g. see [2]). Nevertheless, we emphasize
some points here. We let N denote the set of positive integers {1, 2,
3, . . .} and [n]
S
denote the set {1, 2, . . . , n}. For any set S and k ∈ N, we denote k the set whose
elements are the subsets of S which have k elements. Thus, for |S| = n ∈ N we have
S n
n!
k = k = k!(n − k)! .
Given sequences {an }, {bn } of positive numbers, we write an = O(bn ) if there
exists a constant C ≥ 0 such that |an | ≤ Cbn for all large enough n. And we write
an = o(bn ) or an bn if an /bn → 0 as n → ∞.
1.2. Graphs. A (simple) graph G is a pair (V, E) where E ⊂ V2 . We call V the
set of vertices and E the set of edges of G. Unless otherwise stated, a graph is always finite and simple, and the first subscript of some graph indicates its number
of vertices, e.g., Kn is the complete graph, Cn is the cycle and Pn is the path each with
n vertices. The complete k-partite graph with partition sets of order n1 , . . . , nk is denoted by Kn1 ,...,nk .
The length of a path is the number of its edges and, if x is its first vertex and x0 is
its last vertex, then we call it an (x, x0 )-path. Given a set X of vertices of a graph G,
G[X] denotes the subgraph induced by the edges with both ends in X. Also, G \ X
denotes the subgraph obtained by deleting the vertices of X and the edges incident
to the deleted vertices. Finally Ḡ denotes the complement of G, the graph with same
vertex set of G and whose edge set is the complement of E(G).
RAMSEY THEORY FOR GRAPHS
3
The maximum degree of the vertices of a graph G is denoted by ∆(G). Given two
disjoint non-empty sets of vertices X and Y , we let E(X, Y ) denote the set of all the
edges with one end in X and the other one in Y .
We denote the bipartite subgraph of G with bipartition X ∪ Y and the edge set
E(X, Y ) by G[X, Y ], and in general for disjoint sets X1 , X2 , . . . , Xk we denote by
G[X1 , X2 , . . . , Xk ] the multipartite graph induced by the edges of G from Xi to Xj
for every i 6= j. Furthermore, when there is no risk of confusion, we use G to denote
the multipartite complement of such G, which is defined as the graph we obtain from
the usual complement of G by deleting all edges within the classes in the given
vertex partition.
In some of our theorems/lemmas we use non-standard looking subscripts for an
absolute or relative constant in its statement. We note that these subscripts are equal
to the reference number of the theorem/lemma. This makes it much easier for the
reader to find the place where a constant is defined.
1.3. Ramsey Theorems in Graph Theory. Before we state the most common versions of Ramsey’s Theorem, let us introduce some more notation.
An t-colouring of a set S is any map c : S → [t]. We can also think of a colouring
as a ordered partition S = S 1 ∪ S 2 ∪ . . . S t , where S i is the set of elements of colour
i in S, 1 ≤ i ≤ t. Each set S i is called a colour class. A set is monochromatic if
all its elements have the colour.
Furthermore, given a set S, a positive integer
k
S
T
and a colouring of the set k , we say that a set T ⊂ S is monochromatic if k is
monochromatic.
Whenever we talk about a t-colouring of a graph G we mean a t-colouring of the
edge-set of G.
Given an integer t and graphs G, L1 , . . . , Lt , we say that G arrows (L1 , . . . , Lt ) and
write G → (L1 , . . . , Lt ), if for every t-colouring of G, there exists a colour i, with 1 ≤
i ≤ t, such that the graph induced by the edges of colour i contains Li as a subgraph,
(not necessarily as an induced subgraph). The main question is how large the order
of G must be (in terms of the orders of Li ) to guarantee that G → (L1 , . . . , Lt ). We
define the Ramsey number of the graphs L1 , . . . , Lt , denoted by r(L1 , . . . , Lt ), as the
minimum integer N such that KN → (L1 , . . . , Lt ), if such N exists (and let it be
infinite otherwise).
The most studied and classic question in Ramsey theory arises when we consider
the case in which G and Li , for all 1 ≤ i ≤ t, are complete graphs. So, given integers
`1 , . . . , `t , we would like to determine the value of r(K`1 , . . . , r(K`t ). In that case,
the notation is simplified to r(`1 , . . . , `t ). This is considered a very hard problem.
Even in the simplest case where r = 2 and `1 = `2 , the precise order of magnitude
of r(`, `) is unknown. Those are called diagonal Ramsey numbers and the notation
is usually simplified further to r(`) = r(`, `). Next, we state and prove the most
common version of Ramsey’s theorem.
Theorem 1.1 (Ramsey). For any positive integers `1 , . . . , `t , we have that r(`1 , . . . , `t ) is
finite.
Proof. We will prove the result for t = 2 and leave the general case as an exercise. We
use induction on `1 + `2 . For `1 + `2 = 2 we must have `1 = `2 = 1 and the statement
holds (vacuously), that is r(1, 1) = 1. During the induction step we will also use that
r(`, 1) = r(1, `) = r(`, 2) = r(2, `) = l for any ` ∈ N, which is obvious.
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FABRICIO SIQUEIRA BENEVIDES
Now, assume that we are given `1 , `2 ≥ 2 such that r(`1 , `2 − 1) and r(`1 − 1, `2 ) are
finite. We claim that
(1.1)
r(`1 , `2 ) ≤ r(`1 , `2 − 1) + r(`1 − 1, `2 ).
In fact, let N = r(`1 , `2 −1)+r(`1 −1, `2 ). We only have to check that KN → (`1 , `2 ).
let G = KN and consider any 2-colouring χ of G. Choose any vertex x ∈ V (G) and
define
V1 = {y ∈ V (G) : χ({x, y}) = 1},
V2 = {y ∈ V (G) : χ({x, y}) = 2}.
Therefore V (G) is the disjoint union V1 ∪ V2 ∪ {x} and so |V1 | + |V2 | = N − 1, which
implies
(1) |V1 | ≥ r(`1 − 1, `2 ) or
(2) |V2 | ≥ r(`1 , `2 − 1).
Suppose that (1) occurs. By definition of r(`1 − 1, `2 ), the set V1 either contains a
monochromatic complete subgraph of colour 2 of order `2 or contains one of colour
1 of order `1 − 1. In the first case we are already done. In the second case adding x to
such subgraph yields a clique of order `1 of colour 1, so we are also done. The case
in which (2) occurs is analogous.
We remark that the original version of Ramsey’s theorem was for infinite uniform
hypergraphs, as stated bellow.
Theorem 1.2. For any k, t ∈ N, and any colouring of Nk with t colours, there is a infinite
set S ⊂ N such that Sk is monochromatic.
Proof. Note that the case k = 1 is trivial and is equivalent to the (infinite) pigeonhole
principle:
“If you partition an infinite set into a finite number of sets then one of those sets
needs to be infinite.”
Next, we prove the case k = 2 and leave the general case as an exercise for the
reader. The general case follows by induction on k with a similar argument. We will
use the pigeonhole twice during the proof. Consider any t-colouring c : N2 → [t].
For each vertex v ∈ N and colour i ∈ [t], define Ni (v) = {w : c(vw) = i}, the set of
neighbours of v through an edge of colour i.
We first construct a sequence (x1 , x2 , . . .) of vertices such that the colour of the
edge xi xj is determined by the minimum of i and j. Indeed, let x1 = 1 and X1 = N,
and observe that (by the pigeonhole principle) Nt1 (x1 ) is infinite for some t1 ∈ [t].
Let X2 be that infinite set (if more than one choice for t1 then choose one arbitrarily).
Now, given (x1 , x2 , . . . , xn−1 ) and an infinite set Xn , choose xn ∈ Xn arbitrarily, note
that Ntn (xn ) is infinite for some tn ∈ [r], and let Xn be that infinite set. Since the sets
Xn are nested, it follows that c(xi xj ) = ti .
Finally, consider the sequence (t1 , t2 , . . .), and observe that it contains an infinite
monochromatic subsequence. Let (q1 , q2 , . . .) be the indices of this subsequence.
Then (xq1 , xq2 , . . .) is an infinite monochromatic subset of N.
There is also a finite counterpart of Theorem 1.2, which generalises Theorem 1.1.
Theorem 1.3. For any k, `1 , . . . , `t ∈ N, there exists N0 = rk (`1 , . . . , `t ) such that for any
N > N0 and any t-colouring of [Nk ] there is a set A ⊂ [N ] and i, with 1 ≤ i ≤ t and
|A| = `i , such that Ak is monochromatic of colour i.
RAMSEY THEORY FOR GRAPHS
5
1.4. Ramsey Theory outside Graph Theory. As mentioned earlier, Ramsey theory
is not limited to graphs. It appears in many other set-ups. In this section, we state
various theorems in Ramsey Theory which are not directly related to Graph Theory.
We will prove only a couple of them. We encourage the reader to read the remaining
proofs in [9].
Theorem 1.4 (Schur). For every t, there N0 such that for any N > N0 and any t-colouring
of [N ] there are three integers a, b, c of the same colour such that a + b = c.
Proof. Consider any colouring χ : N → [t], and define the colouring χ0 : N2 → [t] by
χ0 ({a, b}) = χ(|a − b|).
By Ramsey’s Theorem (Theorem 1.2), for the given t and with k = 2, there is a
monochromatic triangle {x, y, z}, say with x < y < z (in fact, there is an infinite
monochromatic set). But then c(y−x) = c(z−x) = c(z−y), and since (y−x)+(z−y) =
(z − x) we have the desired result (with a = y − x, b = z − y and c = z − x.
Remark 1.5. One can also also prove Schur’s theorem without the use of Ramsey’s
theorem.
Another very curious consequence of Ramsey’s theorem is the following result
due to Erdős and Szekeres (1935).
Theorem 1.6 (Erdős-Szekeres). For any n ∈ N, there is an integer f (n) such that any set
of at least f (n) points in the plane, where no tree of them are collinear, contains a subset of
order n which form a convex polygon.
Proof. We claim that it is enough to take f (n) = r4 (n, 5). We leave the details of the
proof as an exercise. Hints: prove and use the following facts. a) any set of 5 points
(no tree of then collinear) contains 4 points which form a convex quadrilateral (this
fact is known as the “Happy Ending problem”); b) if n points are such that any 4
form a convex quadrilateral then they form a convex n-gon.
n
Remark 1.7. The best known upper bound is f (n) ≤ 2n−5
+ 1 = O √4 n from [24].
n−2
The next theorem is another early result in Ramsey Theory. It is a important wellknown theorem about finding monochromatic arithmetic progressions.
Theorem 1.8 (van der Waerden, 1927). For every `, t, we have that any t-colouring of N
contains an arithmetic progression of ` terms of the same colour.
The theorem above can also be stated in a finite form.
Theorem 1.9 (van der Waerden, 1927). For every `, t, there exists N0 = W (`, t) such
that for all N > N0 we have that any t-colouring of [N ] contains an arithmetic progression
of ` terms of the same colour.
The next theorem guarantees, in particular, that a (generalised) tic-tac-toe game
played on a board of high enough dimension will never finish in a draw.
Theorem 1.10 (Hales-Jewett). For every t, k, there exists N0 = HJ(k, t) such that for all
N > N0 we that every t-colouring of the N -dimensional cube
{(x1 , . . . , xN ) : xi ∈ [k], 1 ≤ i ≤ k}
contains a monochromatic “line”.
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FABRICIO SIQUEIRA BENEVIDES
We also encourage the reader to look for other important theorems like the Theorem of Rado and the Theorem of Graham-Leeb-Rothschild, to name a few which can
also be found in [9]. We also recommend the fine lecture notes on Ramsey Theory
by Imre Leader [13].
1.5. Bounds on Ramsey numbers. Although it is considered very hard to compute
r(`) precisely, one can easily prove some bounds on it as shown by the next wellknown theorem. Its proof is also very insightful.
Theorem 1.11. We have that 2`/2 ≤ r(`) ≤
22`−2
√ .
`
Proof. We prove the upper bound and the lower bound of Theorem 1.11 separately.
For the upper bound we shall use induction to prove the following slightly better
bound:
k+`
r(k, `) ≤
.
k
It is trivial to check that this is true for small values of k and ` (say if either of them
is equal to 1 or 2). The induction step follows from equation (1.1):
r(k, `) ≤ r(k − 1, `) + r(k, ` − 1) ≤
k+`−3
k+`−3
k+`−2
≤
+
=
.
k−2
k−1
k−1
22`−2
For k = `, we get r(`) ≤ 2`−2
≤ √` .
`−1
The proof of the lower bound is a quintessential example of the most basic application of the probabilistic method. The goal is to prove that there is a colouring of
Kn , where n = 2`/2 , which does not contain a monochromatic clique of order `. We
colour the edges of Kn randomly, so that every edge is coloured independently and
receives colour i with probability 1/2 for i = 1, 2. Let X be the number of cliques
of order ` which are monochromatic. The expected value of X is, by linearity of
expectation, equal to the number of sets of order ` times the probability of one such
set being monochromatic, that is:
n 2
2n`
21+`/2
<
=
< 1.
` 2(n`)
`!2`(`−1)/2
`!
Since the expected value of X is less than 1 and X is an non-negative integer, there
must be a colouring for which X = 0 as desired.
Remark 1.12. It turns our to be quite difficult to construct concrete examples of
colourings with no large monochromatic cliques, although nowadays some are known.
It is hard to provide any substantial improvement on these bounds. The best
known bounds are
log(k−1)
2`2`/2
−C log log(k−1) 2` − 2
√ ≤ r(`) ≤ k
,
`−1
e 2
where C is a constant. The lower bound is due to Spencer (1975) and the upper
bound to David Conlon (2009) [5].
RAMSEY THEORY FOR GRAPHS
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1.6. Density Theorems (Turán/Szemerédi). In this section, we consider a different
kind of problem. Let us first define what we mean by density, in two different con
texts. The density of a graph Gn on n vertices is defined as d(Gn ) = |E(Gn )|/ n2 . A
graph Gn is called γ-dense if it has density at least γ.
For a set of natural numbers S ⊂ [n] we define the density of S in [n] by |S|/n.
And finally, for an infinity set A ⊂ N, we define the upper-density of A as:
|A ∩ [n]|
¯
d(A)
= lim sup
.
n
n→∞
In general, a problem in Ramsey theorem could be formulated as follows: given
a number of colours t, we want to determine if every t-colouring of a certain set
contains a monochromatic subset which satisfies a certain property. Now, for a density problem our question is: what is the minimum density γ such that any set with
density larger than γ must satisfies a certain property. In other words, what is the
maximum density that a set may have avoiding to satisfy such property. For example, related to Schur’s theorem, we could ask the following question.
Question 1.2. What is the maximum (upper)-density that a set A ⊂ N may have if
there are no x, y, z ∈ A such that x + y = z.
The answer is simple. The set of odd numbers has density 1/2 and clearly contains
no solution to the equation x + y = z. So, the answer should be at least 1/2. We
claim that we cannot do any better. In fact, consider any ‘sum-free’ set A and let
An = A ∩ [n] and n − An = {n − a : a ∈ An }. If n ∈ An then these two sets are disjoint.
¯
Therefore, An ≤ n/2. From this, it is not hard to conclude that d(A)
≤ n/2.
However, changing the question slightly can make it much harder to answer.
Question 1.3. What is the maximum (upper)-density that a set A ⊂ N may have if
there are no x, y, z ∈ A such that x + y = 2z.
Note that in this case, x, y, z form an arithmetic progression. So, one can think of
the following natural and more general question.
Question 1.4. Given an integer k ≥ 3, what is the maximum (upper) density that a
set A ⊂ N may have if it does not contains an arithmetic progression with k terms.
Note that this question is the natural ‘density problem’ which is related to the van
der Waerden theorem. The answer to Question 1.4 is zero (and so to Question 1.3 as
well)! This is due to the following famous and remarkable theorem of Szemerédi.
¯
Theorem 1.13 (Szemerédi [22]). For any integer `, for any set A ⊆ N with d(A)
> 0, we
have that A contains a (non-trivial) arithmetic progression with at least ` terms.
It is an easy observation that Szemerédi’s theorem is a strong generalisation of
Theorem 1.9 (van der Waerden). In fact, notice that any t-colouring of N must contain a colours class with upper-density at least 1/t. To prove Theorem 1.9 for tcolours, it will be enough to look for an AP in the colour class with density at least
1/t and for that we can just apply Szemerédi’s theorem with ε = 1/t.
However, as we saw by answering Question 1.2 the same kind of generalization
does not apply to Schur’s theorem. That is, in Schur’s result, the fact that the colour
classes partition N is essential, so it is not enough to use that one of the colour classes
is large.
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FABRICIO SIQUEIRA BENEVIDES
Remark 1.14. The theorem of van der Waerden was proved in 1927. In 1936, Erdős
and Turán conjectured Theorem 1.13 above. For ` = 2 the result is trivial. The particular case where ` = 3 was first proved in 1953 by Roth [20]. In 1969, the particular
case where ` = 4 was proved by Szemerédi [21] who also proved the general theorem in 1973. The complete proof is extremely complex and intricate. Nowadays
various different proofs are known, using techniques from many different fields like
Ergodic Theory (in the proof by H. Furstenberg) and Fourier analysis (in the proof
by T. Gowers). In section 3.2, we shall give a proof of Roth’s theorem.
Density problems are also well studied in the context of graphs. Fix a graph H.
For each n define
ext(n, H) = max{|E(G)| : |V (G)| = n, H 6⊆ G}.
Here, we are interested in the behaviour of ext(n, H) when H is fixed and n goes
to infinity (which could also be described in term of the density ext(n, H)/ n2 ). The
first result in this direction is due to Turán and for that reason ext(n, H) is known as
the Turán number of H.
Turán has solved the case where H is a complete graph, say H = K`+1 , and proved
that there is only one graph (up to isomorphism) that has ext(n, K`+1 ) edges and
contains no K`+1 . Such graph is called the Turán graph, which we will denote by
T`,n . It is defined as the unique complete `-partite graph on n vertices such that the
number of vertices in each partition class is as equitable as possible (i.e., the number
of vertices in any two classes differ by at most one). It is easy to check that
1
n
1
(1.5)
1−
− O(`) ≤ |E(T`,n )| ≤ 1 −
n2 .
`
2
`
Theorem 1.15 (Turán). For all integer `, n, with ` ≥ 2 we have ext(n, K`+1 ) = |E(T`,n )|
and the only graph G with ext(n, K`+1 ) edges such that K`+1 6⊆ G is T`,n .
Remark 1.16. One could think of Turán Theorem as the density version of the classical Ramsey theorem for cliques (Theorem 1.1). But similarly to what happens with
Schur’s theorem, the theorems of Ramsey and Turán have no direct relation.
Exercises.
(1) Prove that for any integers `1 , . . . , `r ≥ 2 we have
r
X
r(`1 , . . . , `r ) ≤ 2 +
(r(`1 , . . . , `i−1 , `i − 1, `i+1 , . . . , `r ) − 1)
i=1
(2) Give another proof that r(`1 , . . . , `r ) using only that r(`1 , `2 ) is finite.
(3) Show that r(3, 4) = 9 and r(4) = 18. [Hint: consider the graph on {1, . . . , 17},
where ij is an edge iff |i − j| is a a square (mod 17).]
(4) Let rt (`) = r(`, . . . , `) be the Ramsey number of t copies of K` . Show that
Rt+1 (3) − 1 ≤ (t + 1)(Rt (3) − 1) + 1, and use this to prove that
Rk (3) ≤ ek! + 1.
(5) Prove that any sequence of integers has a monotone subsequence.
(6) Colour N with two colours. Must there exist a monochromatic infinite arithmetic progression?
Xn
(7) Prove Theorem 1.3 for any k. Hint: given a colouring c of k+1
construct
Xn \{xn }
0
an appropriate colouring c of
and let Xn+1 be a monochromatic set
k
with respect to c0 .
RAMSEY THEORY FOR GRAPHS
9
(8) Write a complete proof of Theorem 1.6.
(9) Let C(s) be the smallest n such that every connected graph on n vertices has,
as an induced subgraph, either a complete graph Ks , a star K1,s or a path Ps
of length s. Show that C(s) ≤ r(s)s .
(10) Show that there is an infinite set S of positive integers such that the sum of
any two distinct elements of S has an even number of distinct prime factors.
(11) Does there exist a 2-colouring of the infinite subsets of N with no infinite
monochromatic subset?
(12) Prove that equation (1.5) holds.
(13) Let G be a graph on n vertices with |E(Tr−1,n )| edges. By Turán’s Theorem, G
contains a copy of Kr . Show that G also has a copy of Kr+1 − e, the complete
graph minus an edge.
(14) Determine ext(K1,s ) where K(1, s) is a star.
(15) Determine ext(P4 ) where P4 denotes the path of length four. How about for
a path of length k?
(16) The upper density of an infinite graph G is the infimum of all reals α such
that the finite graphs H ⊂ G with |E(H)| |H|
≥ α have bounded order. Use
2
the Erdős-Stone theorem to show that this number always takes one of the
countably many values: 0, 1, 21 , 32 , 34 , . . ..
(17) Let g(n) be the largest number of edges in a graph G on n vertices with the following property: there is a 2-colouring of the edges of G with no monochromatic triangle.
(a) Show that g(n)/ n2 converges.
(b) Find c such that g(n)/ n2 → c as n → ∞.
2. S ZEMERÉDI ’ S R EGULARITY L EMMA
In this section we introduce Szemerédi’s seminal work, the Regularity Lemma.
Much of modern Extremal Graph Theory rests on this fundamental lemma. This
lemma was first conceived, in a weaker form, to be used in the original proof of
Theorem 1.13. We will define the so called reduced graphs and shall also discuss
about a particular class of lemmas, the so called embedding lemmas. In the next
section we shall show how to use those lemmas in combination with the the Szemerédi’s regularity lemma.
First, we note that for bipartite graphs, we will need to use a slightly different
notion of density. For disjoint subsets X, Y of the vertex set of a graph, we let
d(X, Y ) =
|E(X, Y )|
.
|X||Y |
2.1. The Regularity Lemma for Graphs. Loosely put, Szemerédi’s Regularity Lemma
[23] asserts that every graph of positive edge-density can be approximated by the
union of a bounded number of random-like bipartite graphs. Before we can present
it in a formal and precise form, the concept of ε-regular pair needs to be defined.
Definition 2.1. Let G = (V, E) be a graph and let 0 < ε ≤ 1. We say that a pair (A, B)
of two disjoint subsets of V is ε-regular (with respect to G) if
|d(A0 , B 0 ) − d(A, B)| < ε
holds for any two subsets A0 ⊂ A, B 0 ⊂ B with |A0 | > ε|A|, |B 0 | > ε|B|.
10
FABRICIO SIQUEIRA BENEVIDES
Thus, a pair of disjoint sets is regular if the distribution of the edges of the bipartite
graph determined by them is close to uniform. Let us now prove some properties
about ε-regular pairs, in order to get used to them.
Fact 2.2. Let G = (V, E) be any graph. If (X, Y ) is an ε-regular pair of density d with
respect to G then it is also ε-regular pair with respect to the complement, Ḡ, of G now with
density 1 − d).
Proof. Exercise.
The next property states that large subgraphs of a regular pair are still regular but
with a slightly worse ε.
Fact 2.3. If (A, B) is an ε-regular pair with 0 < ε ≤ 1/2, then for any A0 ⊂ A, B0 ⊂ B
such that |A0 | ≥ |A|/2 and |B0 | ≥ |B|/2, the pair (A0 , B0 ) is a 2ε-regular.
Proof. Take A0 ⊂ A0 and B 0 ⊂ B0 such that |A0 | > 2ε|A0 | and |B 0 | > 2ε|B0 |. This
implies that |A0 | > ε|A| and |B| > ε|B|. Since (A, B) is an ε-regular pair, we have
|d(A0 , B 0 ) − d(A, B)| < ε.
Also, since |A0 | ≥ |A|/2 > ε|A| and |B0 | ≥ |B|/2 > ε|B|, we have
|d(A0 , B0 ) − d(A, B)| < ε.
Therefore
|d(A0 , B 0 ) − d(A0 , B0 )| ≤ |d(A0 , B 0 ) − d(A, B)| + |d(A0 , B0 ) − d(A, B)| < 2ε.
Hence, we conclude that (A0 , B0 ) is a 2ε-regular pair.
The third property says that the degree of most vertices is at least some quantity
that close to its expected value.
Fact 2.4. Let G be a bipartite graph with bipartition V (G) = A ∪ B such that the pair
(A, B) is ε-regular with density d = d(A, B). Then, for any Y ⊂ B such that |Y | > ε|B|,
we have
|{x ∈ A : deg(x, Y ) < (d − ε)|Y |}| ≤ ε|A|.
In particular, all but at most ε|A| vertices v ∈ A satisfy deg(v) ≥ (d − ε)|B|.
Proof. Suppose, for a contradiction, that there exists a set Y ⊂ B such that |Y | > ε|B|
and
|{x ∈ A : deg(x, Y ) < (d − ε)|Y |}| > ε|A|.
Let X = {x ∈ A : deg(x, Y ) ≤ (d − ε)|Y |}. Then
X
|E(X, Y )| =
deg(x, Y ) < (d − ε)|X| · |Y |,
x∈X
and therefore
d(X, Y ) < d − ε,
contradicting the fact that (A, B) is ε-regular.
Remark 2.5. Similarly, one can show that most vertices in A satisfy deg(v) ≤ (d+ε)|B|.
This indicates that most vertices have almost the same degree. However, we will
usually apply Fact 2.4 in the way it was stated above.
RAMSEY THEORY FOR GRAPHS
11
The next lemma asserts that regular pairs with positive density contain very long
paths. It is a slightly stronger version of an assertion by Łuczak [15]. The original
version treats the case where the density γ is equal to 1/4. Recall that the subscript of
an absolute or relative constant in the statement of the lemma is equal to its reference
number. This is only to make it easier for the reader to find the place where this
constant is defined.
Lemma 2.6. For every 0 < γ < 1 and ε, with 0 < ε < γ/20, there exists a constant
n2.6 = n2.6 (γ, ε) such that for every n > n2.6 the following holds. Let G be a bipartite graph
with bipartition V (G) = V1 ∪ V2 such that |V1 |, |V2 | = n. Furthermore, let the pair (V1 , V2 )
be ε-regular with density at least γ. Then, for every integer ` with 1 ≤ ` ≤ n − 2εn/γ, and
for every pair of vertices v 0 ∈ V1 , v 00 ∈ V2 satisfying deg(v 0 ), deg(v 00 ) ≥ γn/2, the graph G
contains a (v 0 , v 00 )-path of length 2` + 1.
Proof. Given γ and ε as in the statement, let n2.6 be such that n2.6 ε > 1. Let v 0 and
v 00 as in the statement of the lemma. The strategy for building our path depends on
range of the value of `.
We first consider the case where 1 ≤ ` < γn/3.
For i = 1, 2, set
Vi− = {v ∈ Vi : deg(v) < γn/2}.
Since γn/2 < (γ − ε)|V(3−i) |, by Fact 2.4, we have |Vi− | ≤ ε|Vi |.
Then, setting
Vi+ = Vi \ Vi− ,
we have that |Vi+ | ≥ (1 − ε)n. Take maximum size sets V̂1 ⊆ V1+ and V̂2 ⊆ V2+
satisfying |V̂1 | = |V̂2 |. It is easy to see that the bipartite subgraph H = G[V̂1 , V̂2 ] has
minimum degree at least γn/2 − εn > γn/3. Therefore, we can greedily construct
a path of length 2` − 2, say P2`−2 = v0 v1 . . . v2`−2 , such that v0 = v 0 and V (P2`−2 ) ⊆
V̂1 ∪ V̂2 \{v 00 }. In fact, first choose v0 = v 0 and, assuming that v0 , . . . , vi−1 were chosen,
take vi to be any of the neighbours of vi−1 in V (H) \ {v0 , . . . , vi−1 } ∪ {v 00 }. Such vertex
vi exists given that deg(vi−1 ) > `, and so deg(vi−1 ) − V (P2`−2 ) ≥ 1. To show that we
can extend P2`−2 to a path of length 2` + 1 ending at v 00 , it is enough to show that G
contains an edge {v2`−1 , v2` } from NH (v2`−2 ) \ (V (P2`−2 ) ∪ {v 00 }) to NH (v 00 ) \ V (P2`−2 ).
More precisely, we would get a path P2`+1 = P2`−2 v2`−1 v2` v 00 , i.e., P2`+1 = v0 . . . v2` v 00 .
Such an edge {v2`−1 , v2` } exists because
|NH (v2`−2 ) \ (V (P2`−2 ) ∪ {v 00 })| ≥ γn/2 − εn − γ/3 − 1 > εn
and, similarly,
|NH (v 00 ) \ V (P2`−2 )| > εn.
The ε-regularity of (V1 , V2 ) implies that the density between these sets cannot be
zero, and we note also that those sets are non-empty as εn > 1.
In the range γn/3 ≤ ` ≤ n − 2εn/γ, we use induction on `. Assume that we have
already constructed a path P2`−1 = v0 v1 . . . v2`−1 , such that v0 = v 0 and v2`−1 = v 00 .
The strategy will be to replace one edge of this path by a path of length 3. We say
that a vertex v ∈ V (P2`−1 ) is ‘good’ if it has at least εn neighbours not in V (P2`−1 ),
that is, |NH (v) \ V (P2`−1 )| ≥ εn; otherwise we call v ‘bad’.
If there exists an i, with 0 ≤ i ≤ 2` − 2, such that the vertices vi ∈ V (P2`−1 ) ∩ V1
and vi+1 ∈ V (P2`−1 ) ∩ V2 are good, then we can proceed as above: by the ε-regularity
of (V1 , V2 ), the density between N (vi ) \ V (P2`−1 ) and N (vi+1 ) \ V (P2`−1 ) cannot be
zero. In this case, there must be w0 , w00 6∈ V (P2`−1 ) such that {vi , w0 }, {w0 , w00 } and
12
FABRICIO SIQUEIRA BENEVIDES
{w00 , vi+1 } are edges of G. Therefore, we have a path v0 v1 . . . vi w0 w00 vi+1 . . . v2`−1 of
length 2` + 1 connecting v 0 to v 00 . It remains to prove that such an i exists.
Denote Y = V2 \ V (P2`−1 ). Recall that |Y | ≥ 2εn/γ > εn. Let X be the set of
vertices of V1 which have degree at most (γ − ε)|Y | in Y . By Fact 2.4, |X| ≤ εn. Since
(γ − ε)|Y | > (γ/2)|Y | ≥ εn,
all bad vertices of V1 belong to X. Therefore there are at most εn bad vertices in V1 .
Similarly, there are at most εn bad vertices in V2 . Since there are ` independent
edges in P2`−1 and at most 2εn < γn
≤ ` bad vertices, the bad vertices cannot cover
3
all edges of P2`−1 . Hence, the desired i exists.
Given a graph G and a real number 0 < ε < 1, suppose that we have a partition
V (G) = V0 ∪ V1 ∪ · · · ∪ Vt satisfying the following properties:
• |V0 | ≤ εn; |V1 | = |V2 |= · · · = |Vt |;
• and all but at most ε 2t pairs (Vi , Vj ), 1 ≤ i < j ≤ t, are ε-regular with respect
to G.
This means that most of the pairs of clusters (Vi , Vj ) have the same order and
satisfy Definition 2.1 with some uniform (small) ε. We call this partition ε-regular
with respect to G. Szemerédi proved [23] that every sufficiently large graph has an εregular partition in which the number of clusters is bounded by a function of ε and is
independent of the number of vertices of G. Its precise statement is as follows. The
proof of the Lemma is not hard and can be found, for example, in [6]. The genius of
Szemerédi was to imagine that such a statement could be true. However, since the
proof is not so short, we will not expose it here. We shall focus on understanding
the statement of the lemma and how to apply it.
Lemma 2.7 (Regularity Lemma). For every ε > 0 and m ∈ N there exist integers N2.7 =
N2.7 (ε, m) and M2.7 = M2.7 (ε, m) such that: for any graph G where |V (G)| ≥ N4.8 , there
is a partition of V (G) into t + 1 sets
V = V0 ∪ V1 ∪ . . . ∪ Vt
which is ε-regular (with respect to G) and such that m ≤ t ≤ M4.8 .
Remark 2.8. The sets Vi in the partition given by this lemma are called clusters. The
existence of the cluster V0 above is only for technical reasons: it allows us to assume
that all the other clusters have the same number of elements. Sometimes alternative
formulations are used; for example, one may assume that V0 = ∅ if we weaken the
condition |Vi | = |Vj | to |Vi | − |Vj | ≤ 1 for all i, j.
Note that Lemma 2.7 is vacuously true unless the graph G to which it is applied has positive edge-density. Indeed, G is trivially “approximated” by a union of
empty bipartite graphs.
2.2. Embeddings Lemmas. The Regularity Lemma has been applied to asymptotically solve a number of problems in extremal graph theory. Perhaps the most important classes of extremal problems are the Turán-type problems and the Ramsey-type
problems. Both involve finding large subgraphs with a particular property inside a
larger graph G. An embedding of a graph H into G is a map from V (H) to V (G)
that preserves adjacency (that is, is a copy of H inside G). We loosely use the term
‘embedding lemma’ to refer to lemmas that guarantee the existence of a embedding
of H onto G whenever H and G satisfy a certain property. It is from this kind of
lemma that the regularity really gets its strength.
RAMSEY THEORY FOR GRAPHS
13
A common type of embedding lemma uses various properties about regular pairs
to guarantee the existence of certain bipartite subgraphs in the graph determined
by the pair. For example, in Lemma 2.6, for any fixed positive density γ, choosing
ε small enough and n large enough, one can find ‘very long’ paths between the sets
of an ε-regular pair of density γ. In a more general set up, if one aims to find a
long path in a given graph G, it would be desirable to apply the Regularity Lemma
to G so that we can find lots of regular pairs, then apply Lemma 2.6 to some of
those pairs and finally try to ‘glue’ these paths together to find a longer path. We
note, however, that the Regularity Lemma does not state anything about the density
between the pairs of clusters. Such densities may differ significantly from one pair
of clusters to another and may be zero for some pairs. Then, we may not be able to
apply Lemma 2.6 to all pairs. On the other hand, if the original graph G is dense
and large enough, many of the pairs of clusters shall also have positive density.
Furthermore, since the number of clusters is bounded, each cluster should also have
a relatively large number of vertices. It turns out that most of the difficulty comes
from ‘glueing’ together those paths between regular pairs. Later, in Lemma 4.11, we
prove that this strategy works under certain conditions on the connections between
the clusters. This discussion motivates the following definition of a reduced graph
which grasps the connections between clusters.
Definition 2.9. Given a graph G, two parameters ε, d > 0 and an ε-regular partition
of V (G) into V0 , . . . , Vt such that |V0 | < εn, we define the reduced graph R = R(γ, ε)
as follows: the vertex set of R is V = {1, . . . , t}, and there is an edge from vertices i
to j if and only if (Vi , Vj ) is ε-regular and has density at least γ.
In most applications of the regularity lemma, one chooses the parameters ε and γ
(along with many others) and construct such a reduced graph. One then uses the
fact that many properties of the reduced graph are inherited by the original graph G.
Lemma 2.11, bellow, whose proof can be found in Diestel [6], is probably the most
well-known embedding property related to the regularity lemma. Though we will
not use such proposition to prove our theorems, we believe it is relevant to mention
it. We shall need the following definition in order to state it.
Definition 2.10. Given a graph R, the graph R(s) is the graph obtained by replacing
each vertex v of R by a set of s vertices and each edge of R by a complete bipartite
graph between its two corresponding sets of s vertices. This is commonly known as
a blow-up of R.
Lemma 2.11 (Blowup lemma). For every γ ∈ (0, 1], ∆ > 1, there exists ε0 > 0 with the
following property. Let Gn be a graph on n vertices, s ∈ N, and let R = r(γ, ε) be a reduced
graph of Gn with ε ≤ ε0 such that every cluster contains at least 2s/γ ∆ vertices. Then any
subgraph H of R(s) whose maximum degree is ∆(H) ≤ ∆, is also a subgraph of Gn .
Exercises.
(1) Prove Fact 2.2.
(2) Consider a partition {V1 , . . . , Vk } of a finite set V . Show that the complete
graph on V has about k − 1 as many edges between different partition sets as
edges inside partition sets.
(3) Prove that the regularity lemma (trivially) holds for sparse graphs, that
is, for
every sequence (Gn )n∈N of graphs Gn of order n such that |E(Gn )|/ n2 → 0.
14
FABRICIO SIQUEIRA BENEVIDES
3. A PPLYING THE REGULARITY LEMMA
3.1. Triangle removal lemma. A simple yet very powerful application of regularity
Lemma is the Triangle Removal Lemma.
Lemma 3.1 (The Triangle Removal Lemma (Ruzsa and Szemerédi, 1976)). For every
ε > 0 there exists a δ > 0 such that the following holds: If G is a graph with at most δn3
triangles, then all the triangles in G can be destroyed by removing at most εn2 edges.
Proof. Let ε be given. Let G be graph, say on n vertices, for which we cannot destroy
all its triangles by removing at most εn2 edges. We want to prove that G has at
least δn3 triangles, for some δ depending only on ε. Let ε0 = ε/4, m = 4/ε and
let N = N2.7 (ε0 , m), M = M2.7 (ε0 , m) be the constants we get from the Regularity
Lemma 2.7. Taking δ < 1/N 3 , we can assume that n > N . So, we can apply the
regularity Lemma to G and get a ε0 -regular partition V (G) = V0 ∪ V1 ∪ . . . ∪ Vt with
m ≤ t ≤ M.
Remove from G all edges with at least one end in V0 , all edges inside the clusters,
all between irregular pairs, and between pairs which have density smaller than ε/2.
One can estimate the number of edges removed and check that there less than εn2
of them. Therefore, the remaining graph G1 has a triangle whose vertices belong to
tree different clusters, say V1 , V2 , V3 , such that any pair of those clusters is ε0 -regular
with density at least ε/2. Let ` = |V1 | = |V2 | = |V3 |. By Fact 2.4, all but at most
ε0 ` = ε`/4 vertices of V1 have at least (ε/2 − ε0 )` = ε`/4 neighbours in V2 . The same
hold replacing V2 by V3 . Therefore, all but at most ε`/2 vertices of V1 have degree
at least ε`/4 in both V2 and V3 . Pick any such u ∈ V1 and let N2 and N3 be the set
of neighbours of u in V2 and V3 respectively. Since (V2 , V3 ) is ε0 -regular with density
ε/2, and |N2 |, |N3 | ≥ ε0 `, by definition, the density d(N2 , N3 ) is at least ε/2 − ε0 = ε/4.
Therefore,
ε
ε3 2
|E(N2 , N3 )| ≥ |N2 ||N3 | ≥ 3 ` ,
4
4
each each in E(N2 , N3 ) yields a triangle in G by attaching the vertex u. Now, varying
u and using that
n
n
` ≥ (1 − ε0 ) ≥
,
t
2M
we conclude that G has at least
ε ε3 2 ε ε3 1
1−
` · 3` ≥ 1 −
n3
2
4
2 43 23 M 3
triangles. So we can take
ε ε 3
δ = 1−
.
2
8M
3.2. A proof of Roth’s theorem. In this section, we use the Triangle Removal Lemma
to obtain a proof of Roth’s theorem (the particular case of Szemerédi’s theorem when
` = 3. This proof was obtained by Solymosi.
¯
Theorem 3.2 (Roth, 1954). For any set A ⊆ N with d(A)
> 0, we have that A contains 3
elements in arithmetic progression.
¯
Proof. Let A be any set with d(A)
> 0. First, we define a few parameter. They are
simply chosen in a way to make our proof work, so do not worry about then right
RAMSEY THEORY FOR GRAPHS
15
¯
now. Take any γ with 0 < γ < d(A).
Define ε = γ/36. For such ε, Lemma 3.1 give us
a δ = δ(ε). Choose n sufficiently large so that |A ∩ [n]| > γn and n > 1/δ.
We build a graph G = (V, E) as follows. Let V be a disjoint union V1 ∪ V2 ∪ V3 such
that, for k = 1, 2, 3, we have that |Vk | = kn. For simplicity, we assume that Vk = [kn].
And let
E = {{x, y} : x ∈ V1 , y ∈ V2 , y − x ∈ A}
∪ {{y, z} : y ∈ V2 , z ∈ V3 , z − y ∈ A}
∪ {{x, z} : x ∈ V1 , z ∈ V3 , (z − x)/2 ∈ A}.
Note that any triangle {x, y, z} in G corresponds to an arithmetic progression
a1 , a2 , a3 ∈ A, where a1 = y − x, a3 = z − y and a2 = (z − x)/2 = (a1 + a3 )/2.
Some of these APs may be trivial, that is, we may have a1 = a2 = a3 . The key
idea is to prove that G has many triangles while only few of them lead to trivial
APs. In fact, a triangle which leads to a trivial the AP (a, a, a) must be of the form
{x, x + a, x + 2a} with x ∈ [n], x + a ∈ [2n], x + 2a ∈ [3n]. Clearly, there are at most
n2 such triangles as there are at most n choices for x and for a. Furthermore those
triangles are determined by any two of its vertices, so they are all edge-disjoint. On
the other hand, for any x ∈ [n] and a ∈ A ∩ [n] we have x + a ≤ 2n and x + 2a ≤ 3n,
so there are at least n|A ∩ [n]| ≥ γn2 of them. Since they are edge-disjoint, we cannot
eliminate all of them by removing less that γn2 = ε · (6n)2 = ε · |V (G)|2 edges. By
Lemma 3.1, G must have at least δ · (6n)3 triangles. We conclude that G has at least
(δ · (6n)3 ) − n2 > 1 triangles that lead to non-trivial AP of length 3 in A.
3.3. Two standard applications in Graph Theory. In this section, we will need the
following definition.
Definition 3.3. Given a graph G, a proper colouring of V (G) is a colouring such that
any two adjacent vertices receive different colours. The chromatic number of a graph
G, denoted χ(G), is the minimum number of colours in a proper colouring of G. In
other words, χ(G) is the minimum number t such that we can partition V (G) into t
independent sets.
Theorem 3.4 (Chvátal, Rödl, Szemerédi, Trotter, 1983). For every positiver integer ∆
there is a constant C such that r(H) ≤ C|V (H)| for all graphs H with ∆(H) ≤ ∆.
Proof. Let us first give an idea of the proof. For a given ∆ ≥ 1, we will define suitable
ε > 0 and C ∈ N so that the following works. Let H be any graph, say on n vertices,
and let N = Cn. Consider any 2-colouring of G = KN . Denote G = G1 ∪ G2 , where
Gk is the graph induce by the edges of colour k, for k = 1, 2. We wish to prove
that H ⊆ G1 or H ⊆ G2 . Consider an ε-regular partition, V0 ∪ V1 ∪ . . . ∪ Vt , of G1 ,
as provided by the Regularity Lemma (Lemma 2.7). Consider the reduced graph
R(0, ε), that is, a graph with vertex set [t] such that ij is an edge if and only if (Vi , Vj )
is ε-regular with respect to G1 (we do not impose any condition to the density of
(Vi , Vj )). Note that R is (1 − ε)-dense. By fact 2.2, (Vi , Vj ) is ε-regular with respect to
G1 if and only if it is ε-regular with respect to G2 , so R is also a reduced graph for
G2 . We 2-colour R, say R = R1 ∪ R2 , letting an edge ij ∈ E(R) receive the colour
which appears in the majority of the edges of G[Vi , Vj ] (we colour ij arbitrarily in
case of draws). In that way, if ij gets colour k (where k = 1, 2) then the density of
the edges in Gk [Vi , Vj ] is at least 1/2.
By Lemma 2.11 (Blowup Lemma), in order to prove that H ⊆ Gk , for some k =
1, 2, it is enough to show that H ⊂ Rk (s) for a suitable s. Now, since R has density
16
FABRICIO SIQUEIRA BENEVIDES
at least (1 − ε) and we can take ε very small, using Turán’s Theorem we can find
a very large clique in R. If we can assure a clique of order r(∆ + 1), the Ramsey
number for K∆ , it follows that R1 or R2 contain K∆+1 . And since for any graph H
we have χ(H) ≤ ∆(H) + 1 ≤ ∆ + 1, then the vertices of H can be partitioned into
at most ∆ + 1 independent sets. Therefore, H ⊂ K∆+1 (s) for any s ≥ α(H) where
α(H) is the size of the maximum independent set of H. In particular, this is true for
s = |V (H)| = n.
For the formal proof, let us choose the parameters and do the formal computations. Let ∆ be given, define γ = 1/2. For such ∆ and γ, Lemma 2.11 give us ε0 .
1
Let m = r(∆ + 1). Let ε < ε0 be small enough so that ε < m−1
− 1t for t = m (and
therefore, for any t ≥ m). Finally, let M = M2.7 (ε, m) be given by the regularity
lemma (Lemma 2.7) and select
2∆+1 M
C=
.
1−ε
By definition of R has t vertices and by equation (1.5) and the choice of ε, we have
that
2
t
1 t2
1
t
|E(R)| ≥ (1 − ε)
= (1 − ε) 1 −
≥ 1−
≥ |E(Tm−1,t )|.
2
t 2
m−1 2
Therefore Km ⊆ R. By definition of m, we have K∆+1 ⊆ Rk for some k with
1 ≤ k ≤ 2. It only remains to check that the condition to apply Lemma 2.11 (Blowup
Lemma) are satisfied, to guarantee the implication H ⊆ Rk (n) ⇒ H ⊆ Gk . In fact,
letting ` = |V1 | = . . . = |Vt | we have that
`≥
N − |V0 |
N − εN
1−ε
2s
≥
= Cn
= 2∆+1 s = ∆+1 .
t
M
M
γ
Remark 3.5. Theorem 3.4 was proved in [4]. The proof we have shown here is from
Diestel [6]. It uses basically the same idea as the original proof, but the original
did not use Lemma 2.11 for the embedding. Instead it used an adhoc argument to
embed H in G vertex by vertex.
The following theorem is a fundamental result in Extremal Graph Theory. It was
proved originally without the use of the Regularity Lemma. However, it follows
very nicely as a application of the Regularity Lemma together with Turán’s theorem.
Theorem 3.6 (Erdős-Stone, 1946). For all integers r ≥ 2 and s ≥ 1, and every γ > 0,
there exists an integer n0 such that every graph with n ≥ n0 vertices and at least
|E(Tr−1,n )| + γn2
edges contains Kr (s) as a subgraph.
Proof. We give only a sketch, leaving the details to the reader. The full proof can also
be found in [6]. Let r ≥ 2, s ≥ 1, γ > 0 be given. We define a suitable ε, n0 and
consider any graph G with n > n0 vertices and such that
(3.1)
|E(G)| ≥ |E(Tr−1,n )| + γn2 .
We apply the Regularity Lemma in order to get a ε-regular partition V0 ∪ V1 ∪ . . . ∪
Vt and construct the reduced graph R(γ, ε). Remove the edges inside the cluster,
between irregular pairs, and between clusters with density less than γ and let G1 be
the graph induced by the remaining edges. As in the proof of the “Triangle Removal
RAMSEY THEORY FOR GRAPHS
17
Lemma”, here we removed at most O(εn2 ). All edges of G1 lie in pairs (Vi , Vj ) which
are ε-regular and have density at least γ (for which ij is an edge of R). Therefore,
|E(G1 )| ≥ |E(R)|(n/t)2 . If the number of edges in R is smaller than |E(Tr−1,t )| (note
that here we have a t not n as before), we can give an upper bound on the number
of edges in G which contradicts equation (3.1). We conclude that |E(R)| is at least
|E(Tr−1,t )|.
To finish the proof, we apply Turán’s theorem (Theorem 1.15) to R, which implies
that Kr ⊆ R. And then we apply Lemma 2.11 (Blowup Lemma) to find a copy of
Kr (s) in G.
Corollary 3.7 (Erdős-Stone, 1946). Let H be an arbitrary graph. Then
1
n
ext(n, H) = 1 −
+ o(1)
.
χ(H)
2
Exercises.
(1) The (6,3)-Theorem (Ruzsa and Szemerédi, 1976). Let A ⊂ P(n) be a set system
such that |A| = 3 for every A ∈ A, and such that for each set B ⊂ [n] with
|B| = 6, we have
{A ∈ A : A ⊂ B} ≤ 2.
Then |A| = o(n2 ).
(2) Let R3 (n) be the size of the largest subset of [n] with no 3-AP, and let f (k, n)
denote the maximum number of edges in a graph on n vertices which is the
union of k induced matchings.
.
(a) Prove that R3 (n) ≤ f (n,5n)
n
Hint: consider a graph with edge set (x + ai , x + 2ai ).
(b) Using Szemerédi’s Regularity Lemma, deduce Roth’s Theorem.
(3) In the next exercise, we shall prove the following theorem of Thomassen.
Theorem
3.8 (Thomassen). If G is a triangle-free graph with minimum degree
1
+ ε |G|, then χ(G) ≤ C, for some constant C = C(ε).
3
Given a Szemerédi partition S
(A0 , A1 , . . . , Ak ) of V (G), and d > 0, consider
the auxiliary partition V (G) = I⊂[k] XI where
XI = {v ∈ V (G) : i ∈ I ⇔ |N (v) ∩ Ai | ≥ d|Vi |}.
(a) Show that if |I| ≥ 2k/3 then XI is empty.
(b) Show that if |I| ≤ 2k/3 then XI is an independent set.
(c) Deduce Thomassen’s Theorem.
[You may assume the following strengthening of the Regularity Lemma: that
the reduced graph R for such G has minimum degree (1/3 + ε/2)|R|]
4. R AMSEY NUMBERS OF CYCLES
It is an immediate consequence of Theorem 1.1 that r(L1 , . . . , Ls ) is finite for any
graphs L1 , . . . , Ls . In order to check this, define `i = |V (Li )| for every 1 ≤ i ≤ s. We
can see that
(4.1)
r(L1 , . . . , Ls ) ≤ r(`1 , . . . , `s ).
In fact, for any s-colouring of KN , where N = r(`1 , . . . , `s ), there exists a color i
whose color class contains K`i as a subgraph. The result follows as Li is a subgraph
of K`i and we do not require it to be an induced subgraph.
18
FABRICIO SIQUEIRA BENEVIDES
A much more interesting fact, however, is that sometimes the left-hand side of
inequality (4.1) is much smaller than its right hand side. In fact, it follows from
Theorem 1.11 that r(`1 , . . . , `s ) is at least exponential in min{`1 , . . . , `s } while for
some classes of graphs, as exemplified bellow, the number r(L1 , . . . , Ls ) is linear
in max{`1 , . . . , `s }.
Here, we are particularly interested in the case where the graphs Li are cycles.
The case where s = 2 and the graphs L1 , L2 are cycles of length n, denoted Cn ,
was raised by Bondy and Erdős [3] in 1973 and it was fully solved by Faudree and
Schelp [8], and independently by Rosta [19]. (For a new short proof see Károlyi and
Rosta [12]). They proved the following (without the use of the Regularity Lemma).
Theorem 4.1. Given integers n ≥ 3, we have


if n = 3 or 4
6,
r(Cn , Cn ) = 2n − 1,
if n is odd, n ≥ 5

3n/2 − 1, if n is even, n ≥ 6.
In the same article, Bondy and Erdős [3] conjectured that if n > 3 is odd then
(4.2)
r(Cn , Cn , Cn ) = 4n − 3.
This conjecture was wide open for some good time (and in fact, it is yet not fully
solved). In 1999, Luczak [15] has proved the following approximate result.
Theorem 4.2 (Łuczak, 1999). r(Cn , Cn , Cn ) = (4 + o(1))n.
After the above theorem was proved, various other more precise results have
been proved using similar techniques. For example, Kohayakawa, Simonovits and
Skokan [10] proved the following.
Theorem 4.3. There exists an n0 such that equation (4.2) holds for every n odd with n > n0 .
The case when n is even differs from the case when n is odd. Benevides and
Skokan [1] proved the following.
Theorem 4.4. There exists an integer n1 such that for every even n > n1 ,
(4.3)
r(Cn , Cn , Cn ) = 2n.
For a general number of colours s, one also has general (but not sharp) bounds on
r(Cn , . . . , Cn ) which are linear in n but exponential in s, by Bondy and Erdős [3] and
recently improved by Łuczak, Simonovits and Skokan [16].
Here, we shall concentrate only a couple of the main aspects which are common
to the proofs of Theorems 4.2, 4.3 and 4.4. We will need the following definition.
Definition 4.5. A matching M is a set edges which are pairwise vertex-disjoint. Let
us denote by nK2 the graph on 2n vertices that has only n edges form a matching.
The size of a matching is the number of edges that it contains.
The first main point is that the Ramsey numbers of matchings are known exactly
(matchings are somewhat easier to deal with than cycles). The following theorem is
known since 1975.
Theorem 4.6 (Cockayne and Lorimer, 1975). Given positive integers n1 , . . . , nt ≥ 1
such that n1 = max{n1 , . . . , nt }, we have that
t
X
r(n1 K2 , . . . , nt K2 ) = n1 + 1 +
(ni − 1).
i=1
RAMSEY THEORY FOR GRAPHS
19
Corollary 4.7. Given a positive integer n we have that
r(nK2 , nK2 , nK2 ) = 4n − 2.
The remarkable thing is that the value of r(nK2 , nK2 , nK2 ) is really close to r(C2n , C2n , C2n ),
even though at first sight, the existence of a monochromatic C2n seems to be less
likely than the existence of nK2 (the later has the same same number of vertices, less
edges and is less ‘structured’).
Since our graphs now are multi-coloured, we need the following generalized version of the Regularity Lemma.
Lemma 4.8 (Regularity Lemma). For every ε > 0 and s, m ∈ N there exist integers
N4.8 = N4.8 (ε, s, m) and M4.8 = M4.8 (ε, s, m) such that: for all graphs G1 , . . . , Gs with the
same vertex set V where |V | ≥ N4.8 , there is a partition of V into t + 1 sets
V = V0 ∪ V1 ∪ . . . ∪ Vt
which is ε-regular with respect to each Gk , 1 ≤ k ≤ s , and such that m ≤ t ≤ M4.8 .
Remark 4.9. The original regularity lemma refers to the case s = 1. The proof for an
arbitrary but fixed number s of graphs is is essentially the same as the proof of the
original one. This version is used, for example, by Erdős, Hajnal, Sós, and Szemerédi
[7], and formulated in a survey by Komlós and Simonovits [11].
4.1. From connected matchings to paths. The proofs of Theorems 4.2, 4.3 and 4.4
use some special kinds of matchings.
Definition 4.10. A connected matching is a matching M such that all the edges of M
are in the same connected component C of G. We say that M is an odd connected
matching, if the component C is not bipartite.
The general idea those proofs consists of applying the Regularity Lemma (as
stated in Lemma 4.8, with s = 3) and considering an appropriate reduced graph and
a colouring of it edges. Then looking for a large connected matching in the reduced
graph. Finding such matching is not that easy (doing that takes a good number of
the pages of the mentioned articles). However, it is still much easier than finding
long cycles. Having found such a matching, one can use the following embedding
lemma that tell us how to obtain from the matching a cycle in the original graph G.
The basic idea behind Lemma 4.11 is that one can use Lemma 2.6 to each edge of the
matching in the reduced graph in order to find relatively long paths in G and afterwards use the fact that the matching is connected in order to glue those long paths
into a cycle (we commented about this idea at the beginning of Subsection 2.2).
Lemma 4.11. Given 0 < η < 1/4, there exists c4.11 = c4.11 (η) > 0, such that for any real
numbers 0 < γ < 1 and 0 < ε < 1 satisfying ε/γ ≤ c4.11 and any natural number t, there
exists n4.11 = n4.11 (η, γ, ε, t) such that the following holds. Let Gn be a graph on n > n4.11
vertices and let Rt = Rt (γ, ε) be a reduced graph of Gn on t vertices. If Rt contains a
connected matching M of size t1 ≥ (1/4 + η)t, then Gn contains an even cycle of order
` for any even ` such that 4t < ` ≤ (1/2 + η)n. If, in addition, M is contained in an
odd component, then Gn also contains also odd cycles of any order ` such that 4t < ` ≤
(1/2 + η)n.
Proof. Let 0 < η < 1/4 be given. Choose c4.11 = η/20 and note that such choice
implies that for any reals 0 < γ < 1, 0 < ε < 1 satisfying ε/γ < c4.11 we have
1
8ε
1
+ 2η
1−
(1 − 2ε) ≥
+η .
2
γ
2
20
FABRICIO SIQUEIRA BENEVIDES
Fix such η, γ, ε, and let t be any natural number. We consider the constant n2.6 (γ/2, 2ε)
obtained when we input γ/2 and 2ε to Lemma 2.6. Let n4.11 be such that
(1 − ε)n4.11
> max{2t + 2n2.6 (γ/2, 2ε), 4t/ε, 32γ −3/2 }.
t
Let Gn be any graph on n > n4.11 vertices and let Rt be a reduced graph as in the
statement of the lemma and let V0 , V1 , . . . , Vt , with |V0 | < εn, be the clusters of the
ε-regular partition determining Rt . Note that for any i 6= 0, we have |Vi | = m and
the m ≥ (1 − ε)n4.11 /t choice of n4.11 implies that m − 2t > m/2.
Let M = {a1 b1 , . . . , at1 bt1 } be a monochromatic connected matching in Rt of size
t1 ≥ (1/4 + η)t. Let K be the monochromatic component of Rt containing M .
First, we show that K has a closed walk of even length which contains all edges
of M . Let T be a spanning tree of K such that E(T ) contains all edges of M (this
can be done via Kruskal’s algorithm, i.e., starting with the edges of K and carefully
adding new edges until we get a spanning tree). Let Weven be the minimal closed
walk containing all the edges of T . Such a walk contains each edge of T exactly
twice, therefore it has an even length. Also, its length must be at most 2t.
In the case where K is an non-bipartite component, we can also find a closed walk
of odd length containing all edges of M . In fact, consider some arbitrary vertex r of
T and look at the levels of T as a rooted tree with root r. In this case, there must exist
an edge xy ∈
/ E(T ), such that x and y are in levels of same parity, i.e., the lengths of
the unique paths from x to r and from y to r in T have the same parity. Therefore,
the unique path Pxy from x to y contained in Weven has even length. We can construct
a walk Wodd by taking Weven and replacing Pxy by the edge xy. It is clear that Wodd is
a closed walk, it has odd length and it contains every edge of M (at least once), as
desired.
Now, consider any ` in the range 4t < ` ≤ (1/2 + η)n. We aim to build a C` in
G. We start by letting L = Wodd in the case ` is odd and L = Weven in the case ` is
even. In particular, we can proceed with the case where ` is odd only when such
Wodd exists, i.e., when the component K is non-bipartite. Denote L = i1 i2 . . . is i1 ,
which implies that s and ` have the same parity. Next we use standard regularity
arguments and Lemma 2.6 to build the desired cycle in Gn .
For each j, with 0 ≤ j ≤ s, we say that a vertex in Vij is ‘good’ if it has at least
(γ − ε)|Vij | = (γ − ε)m neighbours in each of Vij−1 and Vij+1 , where we set Vi0 = Vis
and Vis+1 = Vi1 ; and we say that a vertex is ‘bad’ otherwise. Note that for any j, by
Fact 2.4 applied to (Vij , Vij+1 ) and to (Vij , Vij−1 ), at most 2εm vertices of Vij are bad.
The next important step in the proof is to construct a (small) cycle C̃ = vi1 vi2 . . . vis
with vij ∈ Vij such that all its vertices are good. We emphasize that while we may
have Vik = Vij , for some numbers k, j with k 6= j, the vertices vij of C are chosen to
be pairwise distinct. Let us construct such cycle step by step, adding one vertex at
each step. At the first step, we let vi1 be any good vertex in Vi1 (which exists since
(1 − 2ε)|Vi1 | ≥ 1). Suppose that for some j, with 1 ≤ j ≤ s − 3, we have constructed
a path Pj = vi1 vi2 . . . vij in which all vertices are good. In particular, vij has at least
(γ − ε)m neighbours in Vij+1 . Among those, at most 2εm are bad and less than j are
in Pj . Therefore, vij has at least (γ − 3ε)m − j good neighbours not in Pj . Finally,
since j ≤ s ≤ t < γm/2 and 3ε < γ/4, we have (γ − 3ε)|Vij+1 | − j ≥ γ|Vij+1 |/4 ≥ 1.
So there exists vij+1 ∈ |Vij+1 | such that vij+1 is good and vi1 vi2 . . . vij vij+1 is a path. At
step s − 2, we have constructed a path Ps−2 = vi1 vi2 . . . vis−2 in which all vertices are
good. By the same argument as before, vs−2 has at least |Vis−1 |/4 good neighbours
RAMSEY THEORY FOR GRAPHS
21
in Vis−1 but not in Ps−2 ; let A be the set of such neighbours. Similarly, v1 has at
least γ|Vis |/4 good neighbours in Vis but not in Ps−2 ; let B be set of such neighbours.
Because the pair (Vis−1 , Vis ) is ε-regular and |A|, |B| ≥ εm, it follows that G[A, B]
has density at least (γ − ε) > γ/2. Therefore, the number of edges in G[A, B] is at
least γ|A||B|/2 ≥ γ 3 m2 /32 ≥ 1, where the last inequality follows by the choice of
n4.11 . Letting vis−1 vis be any edge of G[A, B], we have that vi1 vi2 . . . vis−1 vis is a cycle
as desired.
For each ak bk ∈ M , we take maximum size sets Va0k ⊂ (Vak \ C̃) ∪ {vak }, Vb0k ⊂
(Vbk \ C̃) ∪ {vbk } satisfying |Va0k | = |Vb0k | and notice that the assumptions of the lemma
give
γ
(4.4)
|Va0k | = |Vb0k | ≥ |Vak | − |C̃| ≥ |Vak | − 2t > n2.6 ( , 2ε)
2
We also note that
deg(vak , Vb0k ) ≥ deg(vak , Vbk ) − t ≥ (γ − ε)|Vbk | − t ≥ γ|Vbk |/2,
(4.5)
where the last inequality follow from the fact that ε < γ/4 and t/m < γ/4 (by the
definitions of ε and n4.11 respectively). Of course, the analogous inequality holds for
deg(vbk , Va0k ).
We can use Lemma 2.6 to replace the edges of C̃ corresponding to edges of M by
long paths resulting in a larger cycle in Gn . Next, we give bound on how large such
cycles can be.
It is clear that |Va0k | ≥ |Vak |/2 and |Vb0k | ≥ |Vbk |/2, which implies that G[Va0k , Vb0k ]
is (2ε)-regular by Fact 2.3. It is also easy to see that G[Va0k , Vb0k ] has density at least
γ − ε > γ/2. By Equations (4.4) and (4.5), together with the fact that 2ε < γ/2
, we
20
are allowed to apply Lemma 2.6 to G[Va0k , Vb0k ] with parameters γ/2 and 2ε: For each
edge ak bk of M , we choose a natural number `k satisfying
1 ≤ `k ≤ (1 − 8ε/γ) min |Vak | − 2t, |Vbk | − 2t ≤ (1 − 8ε/γ) min |Va0k |, |Vb0k | ,
and for any such choice there exists a path Pak ,bk of length 2`k + 1 starting at vak ,
ending at vbk , and consisting only of edges in G[Va0k , Vb0k ]. If we replace the edge vak vbk
tP
tP
1 −1
1 −1
in C̃ by the path Pak ,bk , we get a cycle of order s − t1 +
(2`k + 1) = s +
2`k . So,
k=0
k=0
the length of the expanded cycle can attain any value which has the same parity of
s and is between s + 2t1 and
s+
tX
1 −1
2 (1 − 8ε/γ) min{|Vak | − 2t, |Vbk | − 2t}.
i=0
Furthermore, s + 2t1 < 4t and
s+
tX
1 −1
2 (1 − 8ε/γ) min{|Vak | − 2t, |Vbk | − 2t} ≥
i=0
(1 − ε)n
≥ 2t1 (1 − 8ε/γ)
− 2t
t
1
(1 − 2ε)n
1
≥
+ 2η t(1 − 8ε/γ)
≥
+ η n.
2
t
2
Therefore, the expanded cycle can attain length ` as desired.
22
FABRICIO SIQUEIRA BENEVIDES
5. F UTHER READING AND ACKNOWLEDGEMENTS
Some of the proofs founds here were based on lecture notes of Yuri Lima [14]
(about the Szemerédi Regularity Lemma and its application) and others on notes of
Robert Morris and Roberto Imbuzeiro (about Extremal Graph Combinatorics). We
highly recommend for the reader to check those notes.
For a more general course on Ramsey Theory we recommend reading the lectures
notes on Ramsey Theory of Imre Leader’s [13].
A large survey which contains most the known results about Ramsey numbers
for graphs was written by Radziszowski and is constantly updated by him [17].
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23.
U NIVERSIDADE F EDERAL DO C EARÁ .
E-mail address: [email protected]