On two problems in graph Ramsey theory

0.22 MB·

by  David Conlon

On two problems in graph Ramsey theory
David Conlon Jacob Fox Benny Sudakov
∗ † ‡
0
1
0 Abstract
2
We study two classical problems in graph Ramsey theory, that of determining the Ramsey
n
a number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph
J with a given number of vertices.

0 The Ramsey number r(H) of a graph H is the least positive integer N such that every two-
3
coloring of the edges of the complete graph KN contains a monochromatic copy of H. A famous

result of Chv´atal, Ro¨dl, Szemer´edi and Trotter states that there exists a constant c(∆) such that
]
O r(H) c(∆)n for every graph H with n vertices and maximum degree ∆. The important open
≤
C question is to determine the constant c(∆). The best results, both due to Graham, Ro¨dl and
. Rucin´ski, state that there are constants c and c′ such that 2c′∆ c(∆) 2c∆log2∆. We improve
h this upper bound, showing that there is a constant c for which c(≤∆) 2c≤∆log∆.
t
a ≤
The induced Ramsey number rind(H) of a graph H is the least positive integer N for which
m
there exists a graph G on N vertices such that every two-coloring of the edges of G contains an
[
induced monochromatic copy of H. Erdo˝s conjectured the existence of a constant c such that,

cn
1 for any graph H on n vertices, rind(H) 2 . We move a step closer to proving this conjecture,
v showing that rind(H) 2cnlogn. This im≤proves upon an earlier result of Kohayakawa,Pr¨omeland
5 ≤
Ro¨dl by a factor of logn in the exponent.
4
0
0
. 1 Introduction
2
0
0 Given a graph H, the Ramsey number r(H) is defined to be the smallest natural number N such
1
: that, in any two-coloring of the edges of KN, there exists a monochromatic copy of H. That these
v
numbers exist was first proven by Ramsey [30] and rediscovered independently by Erd˝os and Szekeres
i
X
[17]. Since their time, and particularly since the 1970s, Ramsey theory has grown into one of the most
r
active areas of research withincombinatorics, overlapping variously withgraph theory, numbertheory,
a
geometry and logic.
The most famous question in the field is that of estimating the Ramsey number r(t) of the complete
graph K on t vertices. However, despite some small improvements [32, 5], the standard estimates,
t
that 2t/2 r(t) 22t, have remained largely unchanged for over sixty years. Unsurprisingly then,
≤ ≤
the field has stretched in different directions. One such direction that has become fundamental in its
own right is that of looking at what happens to the Ramsey number when we are dealing with various
types of sparsegraphs. Another is that of determining induced Ramsey numbers, i.e., proving, for any
given H, that there is a graph G such that any two-coloring of the edges of G contains an induced
∗St John’s College, Cambridge, United Kingdom. E-mail: [email protected]. Research supported by a
Junior Research Fellowship at St John’s College.
†DepartmentofMathematics,Princeton,Princeton,NJ.Email: [email protected]. Researchsupported
by an NSFGraduate Research Fellowship and a Princeton Centennial Fellowship.
‡DepartmentofMathematics,UCLA,LosAngeles,CA90095. Email: [email protected]. Researchsupported
in part by NSFCAREER award DMS-0812005 and bya USA-Israeli BSFgrant.
1
monochromatic copy of H. In this paper, we present a unified approach which allows us to make
improvements to two classical questions in these areas.
In 1975, Burr and Erd˝os [2] posed the problem of showing that every graph H with n vertices and
maximum degree ∆ satisfied r(H) c(∆)n, where the constant c(∆) depends only on ∆. That
≤
this is indeed the case was shown by Chv´atal, Ro¨dl, Szemer´edi and Trotter [4] in one of the earliest
applications of Szemer´edi’s celebrated regularity lemma [34]. Remarkably, this means that for graphs
of fixed maximum degree the Ramsey number only has a linear dependence on the number of vertices.
Unfortunately, because it uses the regularity lemma, the bounds that the original method gives on
c(∆) are (and are necessarily [21]) of tower type in ∆. More precisely, c(∆) works out as being an
exponential tower of 2s with a height that is itself exponential in ∆.
The situation was remedied somewhat by Eaton [11], who proved, using a variant of the regularity
lemma, that the function c(∆) can be taken to be of the form 22c∆. Soon after, Graham, Ro¨dl and
Rucin´ski proved [22], by a beautiful method which avoids any use of the regularity lemma, that there
exists a constant c for which
c(∆) 2c∆log2∆.
≤
For bipartite graphs, they were able to do even better [23], showing that if H is a bipartite graph
with n vertices and maximum degree ∆ then r(H) 2c∆log∆n. They also proved that there are
≤
bipartite graphs with n vertices and maximum degree ∆ for which the Ramsey number is at least
2c′∆n. Recently, Conlon [6] and, independently, Fox and Sudakov [19] have shown how to remove
the log∆ factor in the exponent, achieving an essentially best possible bound of r(H) 2c∆n in the
≤
bipartite case. These results were jointly extended to hypergraphs in [7], after several proofs [8, 9, 29]
using the hypergraph regularity lemma.
Unfortunately, if one tries to use these recent techniques to treat general graphs, the best one seems
to be able to achieve is c(∆) 2c∆2. In this paper we take a different approach, more closely related
≤
to that of Graham, Ro¨dl and Rucin´ski [22]. Improving on their bound, we show that c(∆) 2c∆log∆,
which brings us a step closer to matching the lower bound of 2c′∆. ≤
Theorem 1.1 There exists a constant c such that, for every graph H with n vertices and maximum
degree ∆,
r(H) 2c∆log∆n.
≤
A graph H is said to be an induced subgraph of H if V(H) V(G) and two vertices of H are adjacent
⊂
if and only if they are adjacent in G. The induced Ramsey number r (H) is the smallest natural
ind
number N for which there is a graph G on N vertices such that in every two-coloring of the edges of
G there is an induced monochromatic copy of H. The existence of these numbers was independently
proven by Deuber [10], Erd˝os, Hajnal and Po´sa [16] and Ro¨dl [31]. The bounds that these original
proofs give on r (H) are enormous, but it was conjectured by Erd˝os [13] that the actual values
ind
should be more in line with ordinary Ramsey numbers. More specifically, he conjectured the existence
of a constant c such that every graph H with n vertices satisfies r (H) 2cn. If true, the complete
ind
≤
graph shows that it would be best possible.
In a problem paper, Erd˝os [12] stated that he and Hajnal had proved a bound of the form r (H)
ind
≤
22n1+o(1). This remained the state of the art for some years until Kohayakawa, Pr¨omel and Ro¨dl [25]
proved that there was a constant c such that every graph H on n vertices satisfies r (H) 2cnlog2n.
ind
≤
As in the bounded-degree problem, we remove one of the logarithms in the exponent.
2
Theorem 1.2 There exists a constant c such that every graph H with n vertices satisfies
r (H) 2cnlogn.
ind
≤
Itis worthnotingthat thegraph GthatKohayakawa, Pr¨omelandRo¨dluseintheir proofsis arandom
graph constructed with projective planes. This graph is specifically designed so as to contain many
copies of our target graph H. Recently, Fox and Sudakov [18] showed how to prove the same bounds
as Kohayakawa, Pr¨omel and Ro¨dl using explicit pseudo-random graphs. We will follow a similar path.
A graph is said to be pseudo-random if it imitates some of the properties of a random graph. One
suchrandom-like property,introducedbyThomason[35,36], isthatof havingapproximately thesame
density between any pair of large disjoint vertex sets. More formally, we say that a graph G = (V,E)
is (p,λ)-pseudo-random if, for all subsets A,B of V, the density of edges d(A,B) between A and B
satisfies
λ
d(A,B) p .
| − |≤ A B
| || |
p
The usual random graph G(N,p), where each edge is chosen independently with probability p, is itself
a (p,λ)-pseudo-random graph where λ is on the order of √N. A well-known explicit example, known
to be (1,√N)-pseudo-random, is the Paley graph P . This graph is defined by setting V to be the
2 N
set Z , where N is a prime which is congruent to 1 modulo 4, and taking two vertices x,y V to be
N
∈
adjacent if and only if x y is a quadratic residue. For further information on this and other pseudo-
−
random graphs we refer the reader to [27]. Our next theorem states that, for λ sufficiently small, a
(1,λ)-pseudo-random graph has very strong Ramsey properties. Theorem 1.2 follows by applying this
2
theorem to the particular examples of pseudo-random graphs given above.
Theorem 1.3 There exists a constant c such that, for any n N and any (1,λ)-pseudo-random graph
∈ 2
G on N vertices with λ 2 cnlognN, every graph on n vertices occurs as an induced monochromatic
−
≤
copy in all 2-edge-colorings of G. Moreover, all of these induced monochromatic copies can be found
in the same color.
The theme that unites these two, apparently disparate, questions is the method we employ in our
proofs. A simplified version of this method is the following. In the first color we attempt to find a
large subset in which this color is very dense. If such a set can be found, we can easily embed the
required graph. If, on the other hand, this is not the case, then there is a large subset in which the
edges of the second color are well-distributed. Again, this allows us to prove an embedding lemma.
Such ideas are already explicit in the work of Graham, Ro¨dl and Rucin´ski and, arguably, implicit
in that of Kohayakawa, Pr¨omel and Ro¨dl. The advantage of our method, which extends upon these
ideas, is that it is much more symmetrical between the colors. It is this symmetry which allows us to
drop a log factor in each case.
In the next section, we will prove Theorem 1.1. Section 3 contains the proof of Theorem 1.3. The last
section contains some concluding remarks together with a discussion of a few conjectures and open
problems. Throughout the paper, we systematically omit floor and ceiling signs whenever they are
not crucial for the sake of clarity of presentation. All logarithms, unless otherwise stated, are to the
base 2. We also do not make any serious attempt to optimize absolute constants in our statements
and proofs.
3
2 Ramsey number of bounded-degree graphs
The edge density d(X,Y) between two disjoint vertex subsets X,Y of a graph G is the fraction of
e(X,Y)
pairs (x,y) X Y that are edges of G. That is, d(X,Y) = , where e(X,Y) is the number
∈ × X Y
of edges with one endpoint in X and the other in Y. In a gr|ap||h|G, a vertex subset U is called
bi-(ǫ,ρ)-dense if, for all disjoint pairs A,B U with A, B ǫ U , we have d(A,B) ρ. We call
⊂ | | | | ≥ | | ≥
a graph G bi-(ǫ,ρ)-dense if its vertex set V(G) is bi-(ǫ,ρ)-dense. Trivially, if ǫ ǫ and ρ ρ, then
′ ′
≤ ≥
a bi-(ǫ,ρ)-dense graph is also bi-(ǫ,ρ)-dense. Moreover, if ǫ > 1/2, then every graph is vacuously
′ ′
bi-(ǫ,ρ)-dense as there is no pair of disjoint subsets each with more than half of the vertices.
Before going into theproofof Theorem 1.1, we firstsketch for comparison theoriginal idea of Graham,
Ro¨dl, and Rucinski [22] which gives a weaker bound. We then discuss our proof technique. They
noticed that if a graph G on N vertices is bi-(ǫ,ρ)-dense with ǫ = ρ∆/(∆+1) and N 2ρ ∆(∆+1)n,
−
≥
then G contains every n-vertex graph H of maximum degree ∆. This can be shown by embedding H
onevertexatatime. Inparticular,ifared-blueedge-coloringofK doesnotcontainamonochromatic
N
copy of H, then the red graph is not bi-(ǫ,ρ)-dense, and there are disjoint vertex subsets A and B
with A, B ǫN such that the red density between them at most ρ. It is then possible to iterate,
| | | | ≥
at the expense of another factor in the exponent of roughly log(1/ρ), to get a subset S of size roughly
ǫlog(1/ρ)N with red edge density at most 2ρ inside. Picking ρ= 1 , a simple greedy embedding then
16∆
shows that inside S we can find a blue copy of any graph with at most S /4 vertices and maximum
| |
degree ∆.
To summarize, the proof finds a vertex subset S which is either bi-(ǫ,ρ)-dense in the red graph or is
very dense in the blue graph. In either case, it is easy to find a monochromatic copy of any n-vertex
graph H with maximum degree ∆.
We will instead find a sequence of large vertex subsets S ,...,S such that, in one of the two colors,
1 t
each of the subsets satisfies some bi-density condition and the graph between these subsets is very
dense. The bi-density condition inside each S is roughly the condition which ensures that we can
i
embed any graph on n vertices with maximum degree d , where d +...+d = ∆ t+1. A simple
i 1 t
−
lemmaof Lov´aszguarantees thatwecan partition V(H) = V ... V suchthattheinducedsubgraph
1 t
∪ ∪
of H with vertex set V has maximum degree at most d . Our embedding lemma shows that we can
i i
embed a monochromatic copy of H with the image of V being in S . We now proceed to the details
i i
of the proof.
Definition: A graph on N vertices is (α,β,ρ,∆)-dense if there is a sequence S ,...,S of disjoint
1 t
vertex subsets each of cardinality at least αN and nonnegative integers d ,...,d such that d + +
1 t 1
···
d =∆ t+1, and the following holds:
t
−
for 1 i t, S is bi-(ρ2di,ρ)-dense, and
i
• ≤ ≤
for 1 i < j t, each vertex in S has at least (1 β)S neighbors in S .
i j j
• ≤ ≤ − | |
Note that since d + +d = ∆ t+1 and each d is nonnegative, we must have t ∆+1.
1 t i
··· − ≤
Trivially, if a graph is (α,β ,ρ,∆)-dense and α α, β β, and ∆ ∆, then it is also (α,β,ρ,∆)-
′ ′ ′ ′ ′ ′
≥ ≤ ≥
dense.
We say a red-blue edge-coloring of the complete graph K is (α,β,ρ,∆ ,∆ )-dense if the red graph
N 1 2
is (α,β,ρ,∆ )-dense or the blue graph is (α,β,ρ,∆ )-dense. We say that (α,β,ρ,∆ ,∆ ) is universal
1 2 1 2
if, for every N, every red-blue edge-coloring of K is (α,β,ρ,∆ ,∆ )-dense.
N 1 2
4
Lemma 2.1 If β 4(∆ +1)ρ and (α,β,ρ,∆ ,∆ ) is universal, then (1ρ2∆1α,β,ρ,∆ ,2∆ +1) is
≥ 2 1 2 2 1 2
also universal.
Proof: Consider a red-blue edge-coloring of a complete graph K . If the red graph is bi-(ρ2∆1,ρ)-
N
dense, then, taking t = 1, S = V(K ) and d = ∆ , we see that the red graph is (α,β,ρ,∆ )-dense
1 N 1 1 1
and we are done. So we may suppose that there are disjoint vertex subsets V ,V with V , V
0 1 0 1
| | | | ≥
ρ2∆1N such that the red density between them is less than ρ. Delete from V all vertices in at least
0
2ρV red edges with vertices in V ; the remaining subset V has cardinality at least 1 V 1ρ2∆1N.
| 1| 1 0′ 2| 0|≥ 2
Since (α,β,ρ,∆ ,∆ ) is universal, the coloring restricted to V is (α,β,ρ,∆ ,∆ )-dense. Thus, the
1 2 0′ 1 2
red graph is (α,β,ρ,∆ )-dense (in which case we are again done) or the blue graph is (α,β,ρ,∆ )-
1 2
dense. We may suppose the latter holds, and there are subsets S ,...,S each of cardinality at least
1 t
αV 1ρ2∆1αN and nonnegative integers d ,...,d such that d + +d = ∆ t+1, and the
| 0′| ≥ 2 1 t 1 ··· t 2 −
following holds:
for 1 i t, S is bi-(ρ2di,ρ)-dense, and
i
• ≤ ≤
for 1 i < j t, each vertex in S has at least (1 β)S neighbors in S .
i j j
• ≤ ≤ − | |
Since each vertex in V (and hence in each S ) is in at most 2ρV red edges with vertices in V , there
0′ i | 1| 1
are at most 2ρS V red edges between S and V . For 1 i t, delete from V all vertices in at
i 1 i 1 1
| || | ≤ ≤
least 4(∆ +1)ρS red edges with vertices in S . For any given i, there can be at most 1 V
2 | i| i 2(∆2+1)| 1|
such vertices. Therefore, since t ∆ +1, the set V of remaining vertices has cardinality at least
≤ 2 1′
V t 1 V V /2.
| 1|− · 2(∆2+1)| 1| ≥ | 1|
Since(α,β,ρ,∆ ,∆ )isuniversal,thecoloringrestrictedtoV is(α,β,ρ,∆ ,∆ )-dense. Thus,thered
1 2 1′ 1 2
graphis(α,β,ρ,∆ )-dense(inwhichcasewearedone)orthebluegraphis(α,β,ρ,∆ )-dense. Wemay
1 2
supposethelatter holds,andtherearesubsetsT ,...,T each ofcardinality atleast αV 1ρ2∆1αN
1 u | 1′| ≥ 2
and nonnegative integers e ,...,e such that e + +e = ∆ u+1, and the following holds:
1 u 1 u 2
··· −
for 1 i u, T is bi-(ρ2ei,ρ)-dense, and
i
• ≤ ≤
for 1 i < j u, each vertex in T has at least (1 β)T neighbors in T .
i j j
• ≤ ≤ − | |
Note that e + +e +d + +d = ∆ u+1+∆ t+1= (2∆ +1) (u+t)+1. Moreover,
1 u 1 t 2 2 2
··· ··· − − −
β 4(∆ + 1)ρ, implying that for all 1 i u and all 1 j t every vertex in T has at least
2 i
≥ ≤ ≤ ≤ ≤
(1 β)S neighbors in S . Therefore, the sequence T ,...,T ,S ,...,S implies that the blue graph
j j 1 u 1 t
− | |
is (1ρ2∆1α,β,ρ,2∆ +1)-dense, completing the proof. 2
2 2
By symmetry, the above lemma implies that if β 4(∆ +1)ρ and (α,β,ρ,∆ ,∆ ) is universal, then
1 1 2
≥
(1ρ2∆2α,β,ρ,2∆ +1,∆ ) is also universal.
2 1 2
Asalreadymentioned, ifǫ > 1/2, everygraphGisvacuouslybi-(ǫ,ρ)-dense. Asρ20 = 1 > 1/2, setting
·
t = 1 and S = V(G), we have that every graph G is (α,β,ρ,0)-dense. This shows that (1,2ρ,ρ,0,0)
1
is universal, which is the base case h= 0 in the induction proof of the next lemma.
Lemma 2.2 Let h be a nonnegative integer and D := 2h 1. Then (2 2hρ6D 4h,2(D+1)ρ,ρ,D,D)
− −
−
is universal.
5
Proof: As mentioned above, the proof is by induction on h, and the base case h = 0 is satisfied.
Supposeit is satisfied for h, and we wish to show it for h+1. Let D = 2h 1, D = 2D+1 = 2h+1 1,
′
− −
and β = 4(D + 1)ρ = 2(D + 1)ρ 2(D + 1)ρ. Recall that, for β β , if (α,β ,ρ,∆ ,∆ ) is
′ ′ ′ 1 2
≥ ≥
universal then so is (α,β,ρ,∆ ,∆ ). Therefore, since (2 2hρ6D 4h,2(D + 1)ρ,ρ,D,D) is universal,
1 2 − −
(2 2hρ6D 4h,β,ρ,D,D) is also. ApplyingLemma 2.1, we have that (1ρ2D2 2hρ6D 4h,β,ρ,D,2D+1)
− − 2 − −
is universal. Applying the symmetric version of Lemma 2.1 mentioned above, we have that
1 1
ρ2(2D+1) ρ2D2 2hρ6D 4h,β,ρ,2D+1,2D+1 = (2 2(h+1)ρ6D′ 4(h+1),β,ρ,D ,D ),
− − − − ′ ′
2 2
(cid:16) (cid:17)
is universal, which completes the proof by induction. 2
We will use the following lemma of Lov´asz [28].
Lemma 2.3 If H has maximum degree ∆ and d ,...,d are nonnegative integers satisfying d + +
1 t 1
···
d = ∆ t+1, then there is a partition V(H) = V ... V such that for 1 i t, the induced
t 1 t
− ∪ ∪ ≤ ≤
subgraph of H with vertex set V has maximum degree at most d .
i i
The next simple lemma shows that in a bi-(ǫ,ρ)-dense graph, for any large vertex subset B, there are
few vertices with few neighbors in B.
Lemma 2.4 If G is a bi-(ǫ,ρ)-dense graph on n vertices with ǫ 1/n and B V(G) with B 2ǫn,
≥ ⊂ | | ≥
then there are less than 3ǫn vertices in G with fewer than ρ B neighbors in B.
2| |
Proof: Supposefor contradiction that the set A of vertices in G with fewer than ρ B neighbors in B
2| |
satisfies A 3ǫn. Partition A B = C C with C C into two setsof sizeas equalaspossible.
1 2 1 2
| | ≥ ∩ ∪ | |≤ | |
Then the sets A = A C and B = B C are disjoint, A A/2 ǫn, B B /2 ǫn, the
′ 2 ′ 1 ′ ′
\ \ | | ≥ ⌊| | ⌋ ≥ | | ≥ | | ≥
number of edges between A and B is less than A ρ B , and the edge density between A and B is
′ ′ | ′|2| | ′ ′
less than |AA′′|ρ2B|B′| = ρ2 |BB′| ≤ ρ, contradicting G is bi-(ǫ,ρ)-dense. 2
| || | | |
The following embedding lemma is the last ingredient for the proof of Theorem 1.1.
Lemma 2.5 If ρ 1/30 and G isa graph on N 4(2/ρ)2∆α 1n vertices which is(α, 1 ,ρ,∆)-dense,
≤ ≥ − 2∆
then G contains every graph H on n vertices with maximum degree at most ∆.
Proof: Since G is (α, 1 ,ρ,∆)-dense, there is a sequence S ,...,S of disjoint vertex subsets each of
2∆ 1 t
cardinality at least αN and nonnegative integers d ,...,d such that d + +d = ∆ t+1, and
1 t 1 t
··· −
the following holds:
for 1 i t, S is bi-(ρ2di,ρ)-dense, and
i
• ≤ ≤
for 1 i < j t, each vertex in S has at least (1 1 )S neighbors in S .
• ≤ ≤ i − 2∆ | j| j
By Lemma 2.3, there is a vertex partition V(H) = V ... V such that the maximum degree of the
1 t
∪ ∪
induced subgraph of H with vertex set V is at most d for 1 i t. Let v ,...,v be an ordering of
i i 1 n
≤ ≤
the vertices in V(H) such that the vertices in V come before the vertices in V for i< j. Let N(h,k)
i j
denote the set of neighbors v of v with i h. For v V , let M(h,k) denote the set of neighbors
i k k j
≤ ∈
6
v V of v with i h, that is, M(h,k) = N(h,k) V . Notice that M(h,k) d for v V since
i j k j j k j
∈ ≤ ∩ | | ≤ ∈
the induced subgraph of H with vertex set V has maximum degree at most d .
j j
We will find an embedding f : V(H) V(G) of H in G such that f(V ) S for each i. We will
i i
→ ⊂
embed the vertices in increasing order of their indices. The embedding will have the property that
after embeddingthefirsthvertices, ifk > handv V , thentheset S(h,k) of vertices in S adjacent
k j j
∈
to all vertices in f(N(h,k)) has cardinality at least 1(ρ/2)M(h,k) S . Notice that this condition is
2 | || j|
trivially satisfied when h = 0. Suppose that this condition is satisfied after embedding the first h
vertices. The set S(h,k) are the potential vertices in which to embed v after the first h vertices have
k
been embedded, though this set may already contain embedded vertices.
Let j be such that v V . We need to find a vertex in S(h,h+1) to embed the copy of v . We
h+1 j h+1
∈
have
1 1
S(h,h+1) (ρ/2)M(h,h+1) S (ρ/2)dj S
| | j j
| | ≥ 2 | |≥ 2 | |
since M(h,h + 1) d . If d = 0, we may pick f(v ) to be any element of the set S(h,h +
j j h+1
| | ≤
1) f(v ),...,f(v ) . We may assume, therefore, that 1 d ∆. In this case we know, for each of
1 h j
\{ } ≤ ≤
the at most d neighbors v of v with k > h+1 that are in V , that the set S(h,k) has cardinality
j k h+1 j
at least 1(ρ/2)dj S . Let ǫ = ρ2dj. Since, for 1 d ∆ and ρ 1/30, S is bi-(ρ2dj,ρ)-dense,
2 | j| ≤ j ≤ ≤ j
S(h,k) 1(ρ/2)dj S 2ρ2dj S = 2ǫ S and ǫ S = ρ2dj S ρ2∆αN 1, we may apply
| | ≥ 2 | j| ≥ | j| | j| | j| | j| ≥ ≥
Lemma 2.4 in S with B = S(h,k). Therefore, for each vertex v V ,k > h+1 adjacent to v ,
j k j h+1
∈
at most 3ρ2dj S vertices in S have fewer than ρ S(h,k) neighbors in S(h,k). Thus, all but at most
| j| j 2| |
d 3ρ2dj S vertices in S have at least ρ S(h,k) neighbors in S(h,k) for all v V ,k > h+1 that
j · | j| j 2| | k ∈ j
are neighbors of v . Since, for ρ 1/30, we have d 3ρ2dj 1(ρ/2)dj, there are at least
h+1 ≤ j · ≤ 4
1 1
S(h,h+1) d 3ρ2dj S h (ρ/2)dj S d 3ρ2dj S h (ρ/2)dj S h
j j j j j j
| |− · | |− ≥ 2 | |− · | |− ≥ 4 | |−
1
(ρ/2)∆αN h (2/ρ)∆n h >0
≥ 4 − ≥ −
such vertices that are not already embedded. We can pick any of these vertices to be f(v ). To
h+1
continue, it remains to check that any such choice preserves the properties of our embedding. Indeed,
for any k < h+1 for which v is adjacent to v , f(v ) is adjacent to f(v );
h+1 k h+1 k
•
if k > h+1 and v and v are not adjacent, then S(h+1,k) = S(h,k) and M(h+1,k) =
k h+1
•
M(h,k);
if, for some k > h+1, v and v are adjacent and v V with ℓ = j, then M(h+1,k) = 0
k h+1 k ℓ
• ∈ 6
since vertices of V are embedded before vertices of V , ℓ > j, so no vertex of V was embedded
j ℓ ℓ
yet. Also, S(h+1,k) 1 S since N(h+1,k) ∆, the vertices in f(N(h+1,k)) each have
| | ≥ 2| ℓ| | | ≤
at least (1 1 )S neighbors in S , and hence S(h+1,k) S ∆ 1 S = 1 S ;
− 2∆ | ℓ| ℓ | | ≥ | ℓ|− · 2∆| ℓ| 2| ℓ|
if k > h+1, v and v are adjacent and v V , then M(h+1,k) = M(h,k) +1. Moreover,
k h+1 k j
• ∈ | | | |
by our choice of the vertex f(v ), it has at least ρ S(h,k) neighbors in S(h,k). Therefore
h+1 2| |
S(h+1,k) ρ S(h,k) 1(ρ/2)M(h,k)+1 S = 1(ρ/2)M(h+1,k) S , as required.
| | ≥ 2| | ≥ 2 | | | j| 2 | || j|
As we supposed there is an embedding of the first h vertices with the desired property, the above
four facts imply that there is an embedding of the first h+1 vertices with the desired property. By
induction on h, we find an embedding of H in G. 2
We can now prove the following theorem, which implies Theorem 1.1.
7
Theorem 2.1 For every 2-edge-coloring of K with N = 284∆+2∆32∆n, at least one of the color
N
classes contains a copy of every graph on n vertices with maximum degree ∆ 2.
≥
Proof: Let h be the smallest positive integer such that D := 2h 1 ∆. By the definition of D,
− ≥
∆ D < 2∆. Let ρ= 1 , α = 2 2hρ6D 4h ρ6D, and β = 2(D+1)ρ 1 . Lemma2.2 implies that
≤ 8D2 − − ≥ ≤ 2D
every red-blue coloring of the edges of the complete graph K is (α,β,ρ,D,D)-dense. By Lemma 2.5,
N
since
4(2/ρ)2Dα 1n 4(16D2)2D (8D2)6Dn 4(16(2∆)2)4∆(8(2∆)2)12∆n
−
≤ · ≤
= 22(26∆2)4∆(25∆2)12∆n = 284∆+2∆32∆n= N,
at least one of the color classes contains a copy of every graph on n vertices with maximum degree ∆.
2
3 Induced Ramsey numbers
The goal of this section is to prove Theorem 1.3. We will do this by finding, in any 2-edge-coloring of
thepseudo-randomgraphG, acollection ofvertex subsetsS ,...,S satisfyingcertain conditions. The
1 t
conditions in question are closely related to the notion of density that we applied in the last section.
Now, as then, we demand that the graph of one particular color satisfies a certain bi-density condition
within each S . In addition, we demand that between the different S the other color be sparse. This
i i
may look like a simple rearrangement of the condition from the previous section, but, given that we
are now looking at colorings of a pseudo-random graph G rather than the complete graph K , the
N
condition is more general. Moreover, it is exactly what we need to make our embedding lemma work.
Definition: An edge-coloring of a graph G on N vertices with colors 1 and 2 is (α,β,ρ,f,∆ ,∆ )-
1 2
dense if there is a color q 1,2 , disjoint vertex subsets S ,...,S each of cardinality at least αN
1 t
∈ { }
and nonnegative integers d ,...,d with d + +d = ∆ t+1 such that the following holds:
1 t 1 t q
··· −
for 1 i t, S is bi-(f(ρ,d ),ρ)-dense in the graph of color q, and
i i
• ≤ ≤
for 1 i < j t, each vertex in S is in at most β S edges of color 3 q with vertices in S .
i j j
• ≤ ≤ | | −
We say that(α,β,ρ,f,∆ ,∆ )is universalif, for every graphG, every edge-coloring ofG withcolors 1
1 2
and 2 is (α,β,ρ,f,∆ ,∆ )-dense. Note that the density condition used in the last section corresponds
1 2
to the case when G= K and f(ρ,d )= ρ2di. Essentially the same proofs as Lemmas 2.1 and 2.2 give
N i
the following two more general lemmas. We include the proofs for completeness.
Lemma 3.1 Ifβ 4(∆ +1)ρand(α,β,ρ,f,∆ ,∆ )isuniversal, then(1f(ρ,∆ )α,β,ρ,f,∆ ,2∆ +
≥ 2 1 2 2 1 1 2
1) is also universal.
Proof: Consider an edge-coloring of a graph G with colors 1 and 2. If the graph of color 1 is bi-
(f(ρ,∆ ),ρ)-dense, then, taking, q = 1, t = 1, S = V(G) and d = ∆ , we are done. So we may
1 1 1 1
suppose that there are disjoint vertex subsets V ,V with V , V f(ρ,∆ )N such that the density
0 1 0 1 1
| | | | ≥
of color 1 between them is less than ρ. Delete from V all vertices in at least 2ρV edges of color
0 1
| |
8
1 with vertices in V ; the remaining subset V has cardinality at least 1 V 1f(ρ,∆ )N. Since
1 0′ 2| 0| ≥ 2 1
(α,β,ρ,f,∆ ,∆ ) is universal, the coloring restricted to the induced subgraph of G with vertex set
1 2
V is (α,β,ρ,f,∆ ,∆ )-dense. Thus, there is q 1,2 , disjoint vertex subsets S ,...,S V each
0′ 1 2 ∈ { } 1 t ⊂ 0′
of cardinality at least αV and nonnegative integers d ,...,d with d + +d = ∆ t+1 such
| 0′| 1 t 1 ··· t q −
that the following holds:
for 1 i t, S is bi-(f(ρ,d ),ρ)-dense in the graph of color q, and
i i
• ≤ ≤
for 1 i < j t, each vertex in S is in at most β S edges of color 3 q with vertices in S .
i j j
• ≤ ≤ | | −
If q = 1, we are done. Therefore, we may suppose q = 2.
Since each vertex in V (and hence in each S ) is in at most 2ρV edges of color 1 with vertices in
0′ i | 1|
V , then there are at most 2ρS V edges of color 1 between S and V . For 1 i t, delete from V
1 i 1 i 1 1
| || | ≤ ≤
all vertices in at least 4(∆ +1)ρS edges of color 1 with vertices in S . For any given i, there can
2 i i
| |
be at most 1 V such vertices. Therefore, since t ∆ +1, the set V of remaining vertices has
2(∆2+1)| 1| ≤ 2 1′
cardinality at least V t 1 V V /2.
| 1|− · 2(∆2+1)| 1|≥ | 1|
Since (α,β,ρ,f,∆ ,∆ ) is universal, the coloring restricted to the induced subgraph of G with vertex
1 2
set V is (α,β,ρ,f,∆ ,∆ )-dense. Thus, there is q 1,2 , disjoint vertex subsets T ,...,T V
1′ 1 2 ′ ∈ { } 1 u ⊂ 1′
each of cardinality at least αV and nonnegative integers e ,...,e with e + +e = ∆ u+1
| 1′| 1 u 1 ··· u q′ −
such that the following holds:
for 1 i u, T is bi-(f(ρ,e ),ρ)-dense in the graph of color q , and
i i ′
• ≤ ≤
for 1 i < j u, each vertex in T is in at most β T edges of color 3 q with vertices in T .
i j ′ j
• ≤ ≤ | | −
If q = 1, we are done. Therefore, we may suppose q = 2.
′ ′
Note that e + +e +d + +d = ∆ u+1+∆ t+1= (2∆ +1) (u+t)+1. Moreover,
1 u 1 t 2 2 2
··· ··· − − −
β 4(∆ +1)ρ, implying that for all 1 i u and all 1 j t every vertex in T is in at most
2 i
≥ ≤ ≤ ≤ ≤
β S edges of color 1 with vertices in S . Therefore, the sequence T ,...,T ,S ,...,S implies that
j j 1 u 1 t
| |
the edge-coloring of G is (1f(ρ,∆ )α,β,ρ,f,∆ ,2∆ +1)-dense, completing the proof. 2
2 1 1 2
By symmetry, the above lemma implies that if β 4(∆ +1)ρ and (α,β,ρ,f,∆ ,∆ ) is universal,
1 1 2
≥
then (1f(ρ,∆ )α,β,ρ,f,2∆ +1,∆ ) is also universal.
2 2 1 2
Lemma 3.2 Let h be a nonnegative integer and f be such that f(ρ,0) = 1. Define
h
α = 2 2hf(ρ,0) 1f(ρ,2h 1) 1 f(ρ,2i 1)2.
h − − −
− −
Yi=0
Then (α ,2h+1ρ,ρ,f,2h 1,2h 1) is universal.
h
− −
Proof: The proof is by induction on h. As already mentioned, if ǫ > 1/2, every graph G is vacuously
bi-(ǫ,ρ)-dense. Sinceα = 1 > 1/2, setting t = 1 and S = V(G), we have (1,2ρ,ρ,f,0,0) is universal,
0 1
which is the base case h = 0.
Supposethe lemma is satisfied for h, and we wish to show it for h+1. Let D = 2h 1, D = 2D+1 =
′
−
2h+1 1, and β = 4(D + 1)ρ = 2(D + 1)ρ = 2h+2ρ. Note that, for β β , if (α,β ,ρ,f,∆ ,∆ )
′ ′ ′ 1 2
− ≥
9
is universal then so is (α,β,ρ,f,∆ ,∆ ). Therefore, since (α ,2(D + 1)ρ,ρ,f,D,D) is universal,
1 2 h
(α ,β,ρ,f,D,D) is also. Applying Lemma 3.1, we have that (1f(ρ,D)α ,β,ρ,f,D,2D +1) is uni-
h 2 h
versal. Applying the symmetric version of Lemma 3.1 mentioned above, we have that
1 1
f(ρ,2D+1) f(ρ,D)α ,β,ρ,f,2D+1,2D+1 = (α ,β,ρ,f,D ,D ),
h h+1 ′ ′
2 2
(cid:16) (cid:17)
is universal, which completes the proof by induction. 2
A graph G is n-Ramsey-universal if, in any 2-edge-coloring of G, there are monochromatic induced
copies of every graph on n vertices all of the same color. The following lemma implies Theorem 1.3.
Lemma 3.3 If G is (1/2,λ)-pseudo-random on N vertices with λ 2 140nn 40nN, then G is n-
− −
≤
Ramsey-universal.
The set-up for the proof of this lemma is roughly similar to the one presented in the previous section.
We start with a collection of bi-dense sets, in say blue, such that the density of red edges between
each pair of sets is small. The goal is to embed a blue induced copy of a given graph H on vertices
1,...,n. We embed vertices one at a time, always maintaining large sets in which we may embed later
vertices. Suppose that at step i of our embedding, after v ,v ,...,v are chosen, we have sets V for
1 2 i j,i
j > i corresponding to the possible choices for future v . If the vertices j,ℓ > i are not adjacent, then,
j
by the pseudo-randomness of G, the density of nonedges between any two large sets is roughly 1/2,
and it is therefore easy to guarantee that we can pick v and v so that they are nonadjacent. On
j ℓ
the other hand, if the vertices j,ℓ > i are adjacent, then we need to guarantee that v and v will be
j ℓ
joined by a blue edge. Thus, it would be helpful to ensure that the density of blue edges between V
j,i
and V is not too small. In the bounded-degree case we maintain such a property by exploiting the
ℓ,i
fact that the blue density between any two large sets is large. Here, we do not have this luxury in the
case that V and V are subsets of different bi-dense sets in the collection. It is instead necessary to
j,i ℓ,i
use the fact that the underlying graph G is pseudo-random.
To see how this helps, suppose that we now wish to embed v . This will affect the sets V and V ,
i+1 j,i ℓ,i
resulting in subsets V and V . We would like these subsets to mirror the density properties
j,i+1 ℓ,i+1
between V and V . The way we proceed is to show that using pseudo-randomness we can choose
j,i ℓ,i
v such that the density of red edges between the sets V and V remains small. Since G is
i+1 j,i+1 ℓ,i+1
pseudo-random, the total density between large sets is roughly 1/2 and therefore there will still be
many blue edges between these two sets.
Proof of Lemma 3.3: We split the proof into four steps.
Step 1: We will first choose appropriate constants and prepare G for embedding monochromatic
induced subgraphs.
Any (1/2,λ)-pseudo-random graph on at least two vertices must satisfy λ 1/2. Indeed, letting A
≥
and B be distinct vertex subsets each of cardinality 1, we have
λ
1/2 = d(A,B) 1/2 = λ.
| − | ≤ A B
| || |
p
It follows that N 2140nn40nλ 2138nn40n.
≥ ≥
10