On the size-Ramsey number of cycles

0.17 MB·

by  Ramin Javadi

On the size-Ramsey number of cycles
7
R. Javadia, F. Khoeinia, G.R. Omidia,c,1, A. Pokrovskiyb,2
1
0 aDepartment of Mathematical Sciences, Isfahan University of Technology,
2
Isfahan, 84156-83111,Iran

n bDepartment of Mathematics, ETH Zu¨rich, Switzerland
a
J cSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM),

5 P.O. Box 19395-5746,Tehran, Iran
2
[email protected], [email protected],

] [email protected], [email protected]
O
C
Abstract
.
h For givengraphsG1,...,Gk,the size-Ramseynumber Rˆ(G1,...,Gk) is the small-
t
a est integer m for which there exists a graph H on m edges such that in every k-edge
m
coloring of H with colors 1,...,k, H contains a monochromatic copy of Gi of color i
[ for some 1 i k. We denote Rˆ(G1,...,Gk) by Rˆk(G) when G1 = =Gk =G.
≤ ≤ ···
Haxell, Kohayakawa and L uczak showed that the size Ramsey number of a cycle
1
v Cn is linear inn i.e. Rˆk(Cn) ckn for some constantck. Their proof, is basedonthe
≤
8 regularity lemma of Szemer´edi and so no specific constant ck is known.
4 In this paper, we give various upper bounds for the size-Ramsey numbers of
3 cycles. We give an alternative proof of Rˆk(Cn) ckn, avoiding the use of the
7 ≤
regularity lemma. For two colours, we show that for sufficiently large n we have
0
. Rˆ(Cn,Cn) 106 cn, where c=843 if n is even and c=113482 otherwise.
1 ≤ ×
0
Keywords: Ramsey number, Size Ramsey number, Random graphs, Cycles.
7
1 AMS subject classification: 05C55, 05D10
:
v
Xi 1 Introduction
r
a
For given graphs G ,...,G and a graph H, we say that H is Ramsey for (G ,...,G )
1 k 1 k
and wewriteH (G ,...,G ), if nomatter how onecolors theedges of H with k colors
1 k
−→
1,...,k, there exists a monochromatic copy of G of color i in H, for some 1 i k.
i
≤ ≤
Ramsey’s theorem [20] states that for given graphs G ,...,G , there exists a graph H
1 k
that is Ramsey for (G ,...,G ). Note that, if a graph H is Ramsey for (G ,...,G ) and
1 k 1 k
H is a subgraph of H′, then H′ is also Ramsey for (G ,...,G ). In this view, in order
1 k
to study the collection of graphs which are Ramsey for (G ,...,G ), it suffices to study
1 k
the collection (G ,...,G ) of graphs which are minimal subject to being Ramsey for
1 k
F
(G ,...,G ). These graphs are called Ramsey minimal for (G ,...,G ).
1 k 1 k
Many interesting problems in graph theory concern the study of various parameters
relatedtoRamseyminimalgraphsfor(G ,...,G ). Themostwell-knownandwell-studied
1 k
1ThisresearchispartiallycarriedoutintheIPM-IsfahanBranchandinpartsupportedbyagrantfrom
IPM (No. 95050217).
2This research is partially supported by SNSFgrant 200021-149111..
1
one, is the smallest number of vertices of a graph in (G ,...,G ) which is referred to
1 k
F
as the Ramsey number of (G ,...,G ) and is denoted by R(G ,...,G ). In diagonal
1 k 1 k
case, where G = G = = G , we may write R (G) for R(G ,...,G ). Estimating
1 k k 1 k
···
R(K ) = R (K ) is one of the main open problems in Ramsey theory. Erd˝os [10] and
n 2 n
Erd˝os and Szekeres [12] showed that 2n/2 R(K ) 22n, and despite a lot of work,
n
≤ ≤
there have not been improvements to the exponent. For further results about the Ramsey
numbers of graphs, see [6, 19] and the references therein.
In thispaper,we consideranother well-studied parameter called thesize Ramsey num-
berRˆ(G ,...,G )ofthegivengraphsG ,...,G ,whichisdefinedastheminimumnumber
1 k 1 k
of edges of a graph in (G ,...,G ). The investigation of the size Ramsey numbers of
1 k
F
graphs was initiated by Erd˝os et al. [11] in 1978. Since then, the size Ramsey numbers
of graphs have been studied with particular focus on the case of trees, bounded degree
graphs and sparse graphs. The survey paper due to Faudree and Schelp [13] collects some
results about size Ramsey numbers.
One of the most studied directions in this area is the size-Ramsey number of paths.
In 1983 Beck [5], showed that Rˆ(P ) = Rˆ(P ,P ) < 900n for sufficiently large n, where
n n n
P is a path on n vertices. This verifies the linearity of the size-Ramsey number of paths
n
in terms of the number of vertices and from then, different approaches were attempted
by several authors to reduce the upper bound for Rˆ(P ), see [4, 8, 17]. Most of these
n
approaches are based on the classic models of random graphs. Currently, the best known
upper bound is due to Dudek et al. [9] which proved that Rˆ(P ) 74n, for sufficiently
n
≤
large n.
Inthispaper,weinvestigate thesizeRamseynumberofcycles. ThelinearityofRˆ (C )
k n
(in terms of n) follows from the earlier result by Haxell, Kohayakawa and L uczak [16].
Their proof is based on the regularity lemma and has no specific constant coefficient
given. The following theorem is proved in this paper.
Theorem 1.1. Let n ,n ,...,n be a sequence of sufficiently large integers with t even
1 2 t e
numbers and t odd numbers. Let c= 82 352to−2 81te, n = max(n ,...,n ) and suppose
o 1 t
× ×
that for all i, we have n 2 log(nc) +2. Then
i
≥ ⌈ ⌉
Rˆ(C ,...,C ) (lnc+1)c2n.
n1 nt ≤
The above theorem is proved by showing that an Erd˝os-Renyi random graph with
suitable edge probability is almost surely a Ramsey graph for a collection of cycles. By
considering other random graph models we will give further improvements on the bounds
in Theorem 1.1 (see Theorems 3.2, 3.4, and 3.6.)
Throughoutthe paper, the notations logx and lnx refer to the logarithms to the bases
2 and Euler’s number e, respectively.
2 Cycles versus a complete bipartite graph
In this section, we prove some auxiliary results which will later be used to bound the size
Ramsey numbers of cycles. Specifically, we prove some linear upper bounds (in terms of
the number of vertices) for the Ramsey and bipartite Ramsey numbers of cycles versus a
complete bipartite graph. First, we give some definitions and lemmas.
2
A rooted tree with at most two children for each vertex is called the binary tree. The
depth of a vertex in a binary tree T is the distance from the vertex to the root of T and
the maximum distance of any vertex from the root is called the height of T. If a tree has
only one vertex (the root), the height is zero. A perfect binary tree is a binary tree with
all leaves at the same depth where every internal vertex (non-leaf vertex) has exactly two
children. Now we begin with the following lemma.
Lemma 2.1. For every positive integer n 2, there is a binary tree with n leaves at depth
≥
logn and at most 2n+ logn 2 vertices.
⌈ ⌉ ⌈ ⌉−
Proof. If n = 2t for some t, then clearly the perfect binary tree of height t has exactly n
leaves and 2n 1 vertices and we are done. Now, assume that n = 2t1 + +2tr, where
− ···
r 2 and t > > t 0. For each 1 i r, let T be the perfect binary tree of height
1 r i
≥ ··· ≥ ≤ ≤
t with 2ti+1 1 vertices and 2ti leaves. Now, we construct a binary tree T as follows.
i
−
Consider the vertex disjoint binary trees T ,...,T with roots x ,...,x and a new path
1 r 1 r
P = v ...v and add an edge from v to the root x of T for each 1 i r.
1 t1−tr+1 t1−ti+1 i i ≤ ≤
One can easily check that T is a binary tree rooted at v with n = 2t1+ +2tr leaves at
1
···
depth t +1 and
1
r
V(T) = 2ti+1 r+t t +1 2n+t 1
1 r 1
| | − − ≤ −
i=1
X
vertices. Clearly logn = t + 1 and so T is a binary tree with n leaves and at most
1
⌈ ⌉
2n+ logn 2 vertices.
⌈ ⌉−
We also need the following theorem due to Friedman and Pippenger.
Theorem 2.2. [15] Let n and d be positive integers and let G be a non-empty graph such
that for every X V(G) with X 2n 2, N (X) (d+1)X , where N (X) is the
G G
⊆ | | ≤ − | | ≥ | |
set of all vertices in V(G) which are adjacent (in G) to a vertex in X. Then G contains
every tree with n vertices and maximum degree at most d.
The above theorem is used to prove the following lemma about finding red paths in
a 2-coloured balanced complete bipartite graph. The proof technique of this lemma is
similar to techniques from [2, 18].
Lemma2.3. Forgivenintegersm ,m , letnbeanevenintegerwithm n ( logm +
1 2 2 1
≥ ≥ ⌈ ⌉
logm + 1). Suppose that we have a 2-edge-colored K with colors
⌈ 2⌉ 32m1+49m2,32m1+49m2
red and blue and bipartition classes X and Y which has no blue K . Then there are
m1,m2
X′ X and Y′ Y with X′ = m and Y′ = m such that for every x X′ and y Y′
1 2
⊆ ⊆ | | | | ∈ ∈
there is a red path of length n 1 from x to y.
−
Proof. Assume that the edges of H = K are colored by red and blue and (X,Y) is
|X|,|Y|
the bipartition of H with X = Y = 32m +49m . Let H and H be the subgraphs of
1 2 r b
| | | |
H induced on the red and blue edges, respectively. By our assumption, H is K -free.
b m1,m2
Thus, we have N (S) 32m + 48m , for every S X or Y, with S m . In
| Hr | ≥ 1 2 ⊆ | | ≥ 1
particular, this implies that
N (S) 32m +48m , for every S V(G) with S 2m . (2.1)
| Hr | ≥ 1 2 ⊆ | | ≥ 1
3
This holds since for any such set S, either S X m or S Y m .
1 1
| ∩ | ≥ | ∩ | ≥
Claim 1. There is an induced subgraph G H which satisfies
r
⊆
N (S) 4S , for every S V(G) with S 4m +6m . (2.2)
G 1 2
| | ≥ | | ⊆ | | ≤
Proof of the claim. Let A be the largest subset of V(H ) with N (A) < 4A and A
r | Hr | | | | | ≤
4m +6m .Note thatifthereis nosuchasubsetA, thenG = H hasthedesiredproperty.
1 2 r
Now let G be obtained from H by removing vertices in A. To see (2.2), let S V(G) be
r
⊆
a subset with S 4m +6m . For the contrary, suppose that N (S) < 4S . Then
1 2 G
| | ≤ | | | |
N (S A) = N (S) N (A) N (S) + N (A) < 4S +4A = 4S A.
| Hr ∪ | | Hr ∪ Hr | ≤ | G | | Hr | | | | | | ∪ |
By maximality of A, we have S A > 4m +6m . But then N (S A) < 4S A
| ∪ | 1 2 | Hr ∪ | | ∪ | ≤
32m +48m . This contradicts (2.1).
1 2
Now, let G be the subgraph of H which satisfies (2.2). By using Theorem 2.2, G (and
r
so H ) contains a copy of any tree T with at most 2m +3m +1 vertices and maximum
r 1 2
degree at most 3. Now for i= 1,2, let T bea binary tree with m leaves at depth logm
i i i
⌈ ⌉
and at most 2m + logm 2 vertices (which exists due to Lemma 2.1). Also, let T
i i
⌈ ⌉−
be a tree on at most n + 2m + 2m vertices formed by attaching the roots of T and
1 2 1
T by a path of length n 1 logm logm . Note that T has maximum degree 3
2 1 2
− −⌈ ⌉−⌈ ⌉
with leaves x ,...,x , y ,...,y , where there is a path of length n 1 from x to y
1 m1 1 m2 − i j
for every 1 i m and 1 j m . Also note that since n is even, x ,...,x and
≤ ≤ 1 ≤ ≤ 2 { 1 m1}
y ,...,y are contained in different parts of the bipartition of T. By Theorem 2.2, H
{ 1 m2} r
contains a copy of T. Without loss of generality, we can assume that x ,...,x X
{ 1 m1} ⊆
and y ,...,y Y. The sets X′ = x ,...,x and Y′ = y ,...,y satisfy the
{ 1 m2} ⊆ { 1 m1} { 1 m2}
requirements of the lemma.
Given bipartite graphs G ,...,G , the bipartite Ramsey number BR(G ,...,G ) is
1 k 1 k
defined as the smallest integer b such that for any edge coloring of the complete bipartite
graph K with k colors 1,...,k, there exists a monochromatic copy of G of color i
b,b i
in K , for some 1 i k. In other words, it is the smallest integer b such that
b,b
≤ ≤
K (G ,...,G ). The above lemma can be used to give an upper bound for the
b,b 1 k
→
bipartite Ramsey number of an even cycle versus a complete bipartite graph.
Lemma2.4. Forgivenintegersm ,m , letnbeanevenintegerwithm n ( logm +
1 2 2 1
≥ ≥ ⌈ ⌉
logm +1). Then, BR(C ,K ) 32m +49m .
⌈ 2⌉ n m1,m2 ≤ 1 2
Proof. Let H be a 2-edge-colored complete bipartite graph with parts X and Y such that
X = Y = 32m +49m . Supposethat H contains noblueK . To prove the lemma,
| | | | 1 2 m1,m2
it is enough to show that H contains a red C .
n
By Lemma 2.3 there are X′ X and Y′ Y with X = m and Y = m such that
1 2
⊆ ⊆ | | | |
for every x X′ and y Y′ there is a red path of length n 1 from x to y. Since H has
∈ ∈ −
no blue K , there is a red edge xy for some x X′ and y Y′. Adding this edge to
m1,m2 ∈ ∈
the red path of length n 1 from x to y gives a red cycle of length n as required.
−
The following corollary is an immediate consequence of Lemma 2.4.
4
Corollary 2.5. Let m and n ,...,n be positive integers such that for every 1 i t, n
1 t i
≤ ≤
is even and m n 2 log(81t−1m) +1. Then, BR(C ,...,C ,K ) 81tm.
≥ i ≥ ⌈ ⌉ n1 nt m,m ≤
Proof. We give a proof by induction on t. The case t = 1 follows from Lemma 2.4. Now,
assuming the assertion holds for t < t , we are going to prove it for t = t . To see this,
0 0
consider the graph H = K81t0m,81t0m whose edges are colored by the colors 1,2,...,t0+1
and suppose that there is no copy of C of color i in H for all 1 i t . We show
ni ≤ ≤ 0
that there is a copy of K of color t +1 in H. By the induction hypothesis, we have
m,m 0
BR(C ,...,C ,K ) 81t0m and so there is a copy of K in H whose
n1 nt0−1 81m,81m ≤ 81m,81m
edges are colored by the colors t and t +1. Now using Lemma 2.4, this copy contains a
0 0
copy of C of color t , or a copy of K of color t +1. The earlier case does not hold,
nt0 0 m,m 0
since H has no copy of C of the color t . Hence, there is a copy of K of the color
nt0 0 m,m
t +1 in H and we are done.
0
The proof of the following useful lemma is similar to the proof of Lemma 2.3, so we
omit the proof.
Lemma 2.6. For given integers m ,m , let n be an integer with m n ( logm +
1 2 2 1
≥ ≥ ⌈ ⌉
logm +1). Suppose that we have a 2-edge-colored K with colors red and blue
⌈ 2⌉ 33m1+49m2
which has no blue K . Then there are two disjoint subsets of vertices X and Y of
m1,m2
sizes m and m such that for every x X and y Y there is a red path of length n 1
1 2
∈ ∈ −
from x to y.
Thefollowing result follows from Lemma 2.6 along with thesame argument as weused
in the proof of Lemma 2.4.
Lemma 2.7. Let n,m ,m be positive integers, where m n ( logm + logm +1).
1 2 2 1 2
≥ ≥ ⌈ ⌉ ⌈ ⌉
Then, R(C ,K ) 33m +49m .
n m1,m2 ≤ 1 2
We also need the following lemma.
Lemma 2.8. Let n,m ,m be positive integers, where m n ( logm + logm +2).
1 2 2 1 2
≥ ≥ ⌈ ⌉ ⌈ ⌉
Then
K (C ,K ),
X,Y,Z −→ n m1,m2
where K is a complete 3-partite graph with color classes X,Y,Z of sizes X = Y =
X,Y,Z
| | | |
32m +49m and Z = m +m 1.
1 2 1 2
| | −
Proof. The case when n is even follows from Lemma 2.4 (it just suffices to consider the
subgraph K of K and apply Lemma 2.4). Now, let n be odd. Consider a 2 edge
X,Y X,Y,Z
−
coloring of K and suppose that there is no blue K .
X,Y,Z m1,m2
By Lemma 2.3, there are sets X′ X and Y′ Y with X′ = m and Y′ = m such
1 2
⊆ ⊆ | | | |
that for every x X′ and y Y′ there is a red path of length n 2 from x to y contained
∈ ∈ −
in X Y.
∪
Now,sincethereisnoblueK inthe2-edge-coloredK ,wehave Nr(X′) m ,
m1,m2 X,Z | Z | ≥ 1
whereNr(S)isthesetof allvertices inZ whichhave aneighbourinS intheredsubgraph
Z
of K . Similarly, since there is no blue K in the 2-edge-colored K , we have
X,Y,Z m1,m2 Y,Z
Nr(Y′) m . Therefore,since Z = m +m 1,wehaveNr(X′) Nr(Y′) = . Hence,
| Z | ≥ 2 | | 1 2− Z ∩ Z 6 ∅
there are some vertices x X′, y Y′, and z Nr(x) Nr(y). Now, concatenation of
∈ ∈ ∈ Z ∩ Z
the edges yz and zx and the path of length n 2 from x to y in X Y comprises a red
− ∪
C , as required.
n
5
Let f (m ,m )= 33m +49m and for every t 2, define
1 1 2 1 2
≥
f (m ,m ) = f (f (32m +49m ,m +m 1),32m +49m ). (2.3)
t 1 2 t−1 t−1 1 2 1 2 1 2
−
In the following, we show that f (m ,m ) is an upper bound for the Ramsey number of t
t 1 2
cycles (with some restrictions on their sizes) versus the graph K .
m1,m2
Theorem 2.9. Let m ,m and n ,...,n be positive integers such that m n
1 2 1 t 2 i
≥ ≥
2 log(f (m ,m )) +2 for each 1 i t. Then,
t 1 2
⌈ ⌉ ≤ ≤
R(C ,...,C ,K ) f (m ,m ).
n1 nt m1,m2 ≤ t 1 2
Proof. We give a proof by induction on t. The case t = 1 follows from Lemma 2.7. Now,
assuming correctness of the assertion for t < t , we are going to prove it for t = t .
0 0
Consider the (t + 1)-edge-colored graph H = K with colors 1,2,...,t + 1, where
0 N 0
N = f (m ,m ). We assume that H contains no copy of C of color i for each 1 i t
t0 1 2 ni ≤ ≤ 0
and we show that there is a copy of K of color t +1. By the induction hypothesis,
m1,m2 0
we have R(C ,...,C ,K ) N for N = f (32m +49m ,m +m 1) and
n1 nt0−1 N1,N2 ≤ 1 t0−1 1 2 1 2−
N = 32m +49m . So, there is a copy of 2-edge-colored K with parts X and Y by
2 1 2 N1,N2
colors t and t +1 in K . Now, again by the induction hypothesis, we have
0 0 N
X = N = f (32m +49m ,m +m 1) R(C ,...,C ,K ).
| | 1 t0−1 1 2 1 2− ≥ n1 nt0−1 32m1+49m2,m1+m2−1
Therefore, there is a copy of a 2-edge-colored K by the colors t and
32m1+49m2,m1+m2−1 0
t +1 with parts X′ and X′′ in the induced subgraph of K on X. Thus, the edges of the
0 N
complete 3-partite graph with the color classes Y, X′ and X′′ are colored by the colors t
0
and t +1 and so by Lemma 2.8, there is a copy of K of color t +1 in H and we
0 m1,m2 0
are done.
The following corollary follows from Theorem 2.9 and the fact that f (m ,m ) =
2 1 2
38033m +57379m 1617.
1 2
−
Corollary2.10. Forpositiveintegersm ,m ,n ,n withm n n 2 log(38033m +
1 2 1 2 2 1 2 1
≥ ≥ ≥ ⌈
57379m 1617) +2, we have
2
− ⌉
R(C ,C ,K ) 38033m +57379m 1617.
n1 n2 m1,m2 ≤ 1 2 −
By calculating the function f (m ,m ) and using Theorem 2.9, we can prove the fol-
t 1 2
lowing theorem.
Theorem 2.11. Let t 2 and m ,m and n ,...,n be positive integers such that m
1 2 1 t 2
n 2 log(352t−2(32m≥+49m )) +2 for each 1 i t. Then, ≥
i 1 2
≥ ⌈ ⌉ ≤ ≤
R(C ,...,C ,K ) 352t−2(32m +49m ).
n1 nt m1,m2 ≤ 1 2
Proof. UsingTheorem2.9, itjustsufficestoshowthatf (m ,m ) 352t−2(32m +49m ).
t 1 2 1 2
≤
To see this, let f (m ,m ) = a m +b m +c , where a , b and c are three functions in
t 1 2 t 1 t 2 t t t t
terms of t. From (2.3) one can easily see that for each t 2,
≥
a = 32a2 +a b +32b ,
t t−1 t−1 t−1 t−1
6
b = 49a2 +a b +49b ,
t t−1 t−1 t−1 t−1
and
c = a b +a c +c .
t t−1 t−1 t−1 t−1 t−1
−
Clearly for every i 2 we have a < b < 49/32a and so for every t 2,
i i i
≥ ≥
49
a < 32a2 + a2 +49a 35a2 .
t t−1 32 t−1 t−1 ≤ t−1
Therefore, by induction on t we can see that for every t 2, we have
≥
a 32(352t−2),
t
≤
and hence
b 49(352t−2).
t
≤
On the other hand, again by induction on t, we have c 0. Therefore
t
≤
f (m ,m ) a m +b m 352t−2(32m +49m ).
t 1 2 t 1 t 2 1 2
≤ ≤
With all these results in hand, we can prove the main result of this section, as follows.
Theorem 2.12. Let t and t be respectively the number of even and odd integers in the
e o
sequence (n ,...,n ) and suppose that m n 2( logN +1) for each 1 i t, where
1 t i
N = 82 352to−2 81tem. Then ≥ ≥ ⌈ ⌉ ≤ ≤
× ×
R(C ,...,C ,K ) N.
n1 nt m,m ≤
Proof. The case t = 0 follows from Corollary 2.5. So, let t 1. Also, without loss
o o
≥
of generality, assume that n is odd for all 1 i t . Consider a (t +1)-edge-colored
i o
≤ ≤
K with colors 1,2,...,t + 1. Assume that there is no copy of C of color i for each
N ni
1 i t. Ourgoal istoshowthatthereis acopy of K ofcolor t+1. UsingLemma2.7
m,m
≤ ≤
when to = 1 and Theorem 2.11 when to ≥ 2, we have R(Cn1,...,Cnto,K81tem,81tem) ≤ N
and so there is a copy of K81tem,81tem in KN whose edges are colored by te + 1 colors
t + 1,...,t + 1. Now, Corollary 2.5 implies that BR(C ,...,C ,K ) 81tem
o nto+1 nt m,m ≤
and so there is a copy of Km,m of color t+1 in the (te+1)-edge-colored K81tem,81tem, as
desired.
3 Random graphs and upper bounds
In this section, we will apply the obtained results in Section 2 on random graphs to give
somelinearupperboundsintermsofthenumberofvertices forthesizeRamseynumberof
large cycles. For this purpose, we deploy three random structure models namely random
graphs, random regular graphs and random bipartite graphs. First, let us recall a classic
model of random graphs that will be applied in this section. The binomial random graph
(n,p) is the random graph G with the vertex set [n]:= 1,2,...,n in which every pair
G { }
i,j [n] appears independently as an edge in G with probability p. For two subsets
{ } ⊆
7
of vertices S,T, the number of edges with one end in S and one end in T is denoted by
e(S,T). Recall that an event in a probability space holds asymptotically almost surely (or
a.a.s.) if the probability that it holds tends to 1 as n goes to infinity. To see more about
random graphs we refer the reader to see [4, 1]. We will state some results that hold a.a.s.
and we always assume that n is large enough. The first lemma asserts that there is a
graph on N vertices whose number of edges is linear in terms of N, while it has no large
hole (a pair of disjoint subsets of vertices with no edge between them). It should be noted
that similar ideas have been used in [8, 7] to prove linear upper bounds for size Ramsey
numbers.
Lemma 3.1. Let c R and let d = d(c) be such that
+
∈
(1 2c)ln(1 2c)+2cln(c)+c2d 0. (3.1)
− − ≥
Then, in the graph G (N,d/N), a.a.s. for every two disjoint sets of vertices S and T
∈ G
with S = T = cN, we have e(S,T) 1.
| | | | ≥
Proof. Let S and T with S = T = cN be fixed and let X = X = e(S,T). Clearly,
S,T
| | | |
EX = c2dN > 0 and by the Chernoff bound
Pr(X = 0) = Pr(X 0) exp( EX)= exp c2dN .
≤ ≤ − −
(cid:16) (cid:17)
Thus, by the union bound over all choices of S and T we have
N (1 c)N
Pr (X = 0) − exp c2dN
S,T
  ≤ cN cN −
S[,T (cid:18) (cid:19)(cid:18) (cid:19) (cid:16) (cid:17)
  N!
= exp c2dN .
(cN)!(cN)!((1 2c)N)! −
− (cid:16) (cid:17)
Using Stirling’s formula (x! √2πx(x/e)x) we get
∼
N
(1 2c)2c−1exp c2d
1
− −
Pr (X = 0) .
 S,T  ≤ 2πc√1 2cN  c2c (cid:16) (cid:17)
S,T −
[
 1   
= o(1),
≤ 2πc√1 2cN
−
where the last inequality is due to (3.1). This complete the proof.
Combining Lemma 3.1 and Theorem 2.12, gives some information on the size Ram-
sey numbers of cycles. Roughly speaking, these two facts imply that (N,d/N)
G −→
(C ,...,C ) for sufficiently large N when we have some restrictions on the parameters.
n1 nt
In the following result we use this fact to give a linear upper bound for the size Ramsey
number of large cycles. It is a strengthening of Theorem 1.1.
Theorem 3.2. Let f = 82 352to−2 81te, where t and t are respectively the number
e o
× ×
of even and odd integers in the sequence (n ,...,n ). Also let c = min 95412,f if t = 2
1 t
{ }
and c = f, otherwise. Suppose that n = max n ,...,n and for each 1 i t, we have
1 t
{ } ≤ ≤
n 2 log(nc) +2. Then, for sufficiently large n, we have
i
≥ ⌈ ⌉
Rˆ(C ,...,C ) (lnc+1)c2n.
n1 nt ≤
8
Proof. Let N = nc, d= ((2c−1 1)ln(1 2c−1) 2c−1ln(c−1))/c−2 and G= (N,d/N).
− − − G
By Lemma 3.1, a.a.s. for every two disjoint sets of vertices S and T in V(G) with S =
| |
T = n,wehavee(S,T) 1. Therefore,a.a.s. thecomplementofGdoesnotcontain K
n,n
| | ≥
asasubgraph. Ontheotherhand,theexpectednumberofedgesofGis d N Nd/2and
N 2 ≤
the concentration around the expectation follows immediately from the Chernoff bound.
(cid:0) (cid:1)
Hence, for sufficiently large N, there exists a graph H on N vertices with at most Nd/2
edges whose complement does not contain K as a subgraph. Hence, by Corollary 2.10
n,n
and Theorem 2.12, we have H (C ,...,C ). This means that for sufficiently large
−→ n1 nt
N we have
Nd clnc (c 2)ln(c 2)
Rˆ(C ,...,C ) − − − c2n (lnc+1)c2n,
n1 nt ≤ 2 ≤ 2 ≤
where the last inequality follows by applying the mean value theorem to the function
xlnx.
Based on Theorem 3.2, for sufficiently large n, we have
113484 106n if n is odd,
Rˆ(C ,C ) ×
n n ≤ (2515 106n if n is even.
×
Another probability space that we are interested in is the space of random d-regular
graphs on n vertices with uniform probability distribution. This space is denoted by ,
n,d
G
where d 2 is fixed and n is large enough and even when d is odd. Instead of working
≥
directly in , we use the pairing model (also known as the configuration model) of ran-
n,d
G
dom regular graphs, first introduced by Bollob´as [3], which is described here. Suppose
that dn is even and consider dn points partitioned into n labeled buckets v ,v ,...,v
1 2 n
each of which contains exactly d points. A pairing of these points is a perfect matching
into dn/2 pairs. Given a pairing P, we may construct a d-regular multigraph G(P), with
loops and parallel edges allowed, as follows. The vertices are the buckets v ,v ,...,v ,
1 2 n
and a pair x,y in P corresponds to an edge v v in G(P) if x and y are contained in the
i j
{ }
buckets v and v , respectively. It can be easily seen that the probability that the random
i j
pairing yields a given simple graph is uniform, hence the restriction of the probability
space of random pairing to simple graphs is precisely . Moreover, it is well known
n,d
G
that a random pairing generates a simple graph with probability asymptotic to e−(d2−1)/4,
depending on d. Thus, for fixed d, any event holding a.a.s. over the probability space
of random pairing also holds a.a.s. over the corresponding space . For this reason,
n,d
G
asymptotic results over random pairing model are sufficient for our purposes. For more
information on this model, see, for instance, the survey of Wormald [21].
Now let
1
f(a,c,d) = g(c)+g(d)+g((c 2)d)+ g((c 1 a)d) g(c 2) g(ad)
− 2 − − − − −
1 1
g((c 2 a)d) g((1 a)d) g(cd),
− − − − 2 − − 2
9
where g(x) = xln(x). The following lemma is the counterpart of Lemma 3.1 for random
regular graphs and will be used to give another linear upper bound for the size Ramsey
numbers of large cycles.
Lemma3.3. Letc R andd = d(c)benon-negativerealnumberssuchthatf(a,c,d) 0
+
∈ ≤
for every real number a with 0 a 1. Then, a.a.s. for every two disjoint sets of vertices
≤ ≤
S and T in with S = T = N/c, we have e(S,T) 1.
N,d
G | | | | ≥
Proof. LetG = . Ourgoalistoshowthattheexpectednumberofpairsoftwodisjoint
N,d
G
sets, S and T in G, such that S = T = N/c and e(S,T) = 0 tends to zero as N .
| | | | → ∞
This, together with the first moment principle, implies that a.a.s. no such pair exists.
Let m = N/c and a = a(m) be any function of m such that amd Z and 0 a 1. Let
∈ ≤ ≤
X(a) be the expected number of pairs of two disjoint sets S,T such that S = T = m,
| | | |
e(S,T) = 0, and e(S,V (S T)) = adm. Using the pairing model, it is clear that
\ ∪
N N m Nd 2md md
X(a) = − − (amd)!
m m adm adm
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
M(md amd) M Nd md amd M(Nd),
· − · − −
(cid:16) (cid:17).
where M(i) is the number of perfect matchings on i vertices, that is,
i!
M(i) = .
(i/2)!2i/2
After simplification we get
X(a) = (N)!(md)!(Nd md amd)!(Nd/2)!(Nd 2md)!2amd
− − −
(m!)2(adm)!(N 2m)!(Nd 2md adm)!((md adm)/2)!
· − − − −
"
−1
(Nd)!((Nd md amd)/2)! .
− −
#
Using Stirling’s formula (i! √2πi(i/e)i) and focusing on the exponential part we obtain
∼
X(a) = c ef(a,c,d)m,
m
where c = o(1) and
m
f(a,c,d) = clnc+dlnd+(cd 2d)ln(cd 2d)+(cd d ad)/2ln(cd d ad)
− − − − − −
(c 2)ln(c 2) adln(ad) (cd 2d ad)ln(cd 2d ad)
− − − − − − − − −
(d ad)/2ln(d ad) cd/2ln(cd).
− − − −
Our assumptions imply that f(a,c,d) 0 for any integer adm under consideration. Then
≤
we would get X(a) = o(1) (as adm = O(m)). Thus, the desired property is satisfied
adm
and this completes the proof.
P
With the same assumptions as in Theorem 2.12 and Lemma 3.3 for c= 352to−281te+1
we can conclude that (C ,...,C ) for sufficiently large N. In the following
GN,d −→ n1 nt
theorem we use this fact to give a linear upper bound for the size Ramsey numbers of
large cycles.
10