Page 1

Ž.

Advances in Applied Mathematics 22, 48?59 1999

Article ID aama.1998.0620, available online at http:? ?www.idealibrary.com on

On the Ramsey Problem for Multicolor

Bipartite Graphs

W. A. Carnielli?

Department of Philosophy and Center for Logic and Epistemology,

State Uni?ersity of Campinas, Campinas, SP, Brazil

and

E. L. Monte Carmelo²

Department of Mathematics, State Uni?ersity of Maringa, Maringa, PR, Brazil

ÂÂ

Received September 29, 1997; accepted September 19, 1998

Given i, j positive integers, let K

Ž.

R m, n be the smallest integer a such that for any r-coloring of the edges of K

r

one can always find a monochromatic subgraph isomorphic to K

Ž.

words, if a ? R m, n then every matrix a ? a with entries in 0,1,..., r ? 1

r

always contains a submatrix m ? n or n ? m whose entries are i, 0? i ? r ? 1.

Ž.

We shall prove that Rm, n ? 2

n ? 1 ? 2

2

Ž.

previous results R 2, n ? 4n ? 3 and R 3, n ? 8n ? 5 due to Beineke and

2

Schwenk. Moreover, we find a class of lower bounds based on properties of

orthogonal Latin squares which establishes that lim

Academic Press

denote a bipartite complete graph and let

i, j

a, a

. In other

m, n

?4

mŽ.

Ž

m?1

.

? 1, which generalizes the

2

Ž.?2

R 2,2 r

r

? 1.

? 1999

r??

1. INTRODUCTION

The determination of the values of the celebrated Ramsey function

seems to be a formidable task, having resisted a series of mathematical

and computational attacks for more than seven decades. Indeed, exact

values of the Ramsey function are known for only a few small entries. This

has led to the investigation of several particularizations, variants and

generalizations, giving rise to so-called Ramsey theory cf. 9 and 10 , for

.

example , now a central research topic in finite and infinite combinatorics.

Ž

? ???

*E-mail: carniell@cle.unicamp.br.

²E-mail: carmelo@gauss.dma.uem.br.

48

0196-8858?99 $30.00

Copyright ? 1999 by A cademic Press

All rights of reproduction in any form reserved.

Page 2

RAMSEY PROBLEM

49

In this paper we consider a finite variant of the Ramsey function for

bipartite multicolor graphs.

Given integers r ? 2, 1? m ? n, let us define the Ramsey bipartite

Ž.

function R m, n to be the smallest integer a such that for any r-coloring

r

of the edges of a bipartite complete graph K

monochromatic subgraph isomorphic to the bipartite complete graph K

Ž.Ž

The abbreviation R m, n ? R m, n is used to simplify notation.

2

Ž

??

Several papers for example, 11 and 14

approaches to this variant problem, namely, studying the connections with

the classical Zarankiewicz function, or the connections with covering and

packing problems for graphs, or with Steiner systems and other combinato-

rial designs.

Although we remain in the finite case here, it is interesting to mention

that not only the finite, but also the infinite counterparts of the Ramsey

partition problems have enormous interest for the foundations of mathe-

matics, with far-reaching consequences in set theory and model theory see

? ?

.

1 for an overview . An attempt to study the infinite combinatorial

contents of some Ramsey-type problems was made in 6 , where it is shown

that a new partition principle independent of the axiom of choice can be

consistently introduced into the foundations of set theory.

In the finite case, Beineke and Schwenk 2 defined the bipartite Ramsey

numbers for the special case of two colors: R m, n ? R m, n and proved

the following results:

one can always find a

a, a

.

m, n

.

??.

have considered distinct

Ž

? ?

? ?

Ž.Ž.

2

R 1, n ? 2n ? 1,

Ž.

R 2, n ? 4n ? 3,

Ž.

R 3, n ? 8n ? 5.

Ž.

1a

Ž

Ž

Ž

.

.

.

1b

1c

They also conjectured that

R m, n ? 2mn ? 1 ? 1.

Ž.

2

Ž. Ž .

Ž .

?

Ž.

Irving was able to improve the estimate lc to R 3, n ? 8n ? 7 in a

intricate proof which takes four pages of 14 , at the same time obtaining

Ž.

new upper bounds for R m, n , where 4? m ? n ? 7, and showing the

Ž .

falsity of the proposed equality in 2 by means of the example R 4,4 ?

48? 49.

One of our aims in the present paper is to obtain some lower bounds for

the bipartite Ramsey function and, as a main result, to extend 1b and

Ž.

1c , by proving a general upper bound,

?

Ž.

Ž.

R m, n ? 2mn ? 1 ? 2m?1? 1,

Ž.Ž.

Page 3

CARNIELLI AND MONTE CARMELO

50

for all n ? m ? 2. These results obviously constitute evidence for a

Ž .

weaker version of the conjecture 2 ,

Ž.

Ž .

Although the inequality 3 is valid for m ? 3, the case m ? 4 is still

Ž .

open. Indeed, conjecture 3 gives us, in particular, that R 4,6 ? 81 and

Ž.

R 4,7 ? 97; while the known best limits obtained are: R 4,6 ? 82 and

Ž.

??

R 4,7 ? 98, due to Irving 14 .

In order to proceed to our objective we shall make use of the well-known

Zarankiewicz function, defined as follows.

Given natural numbers 1? m ? n ? a, we define the Zarankiewicz

Ž .

number Za as the minimal cardinality of the set of edges of G in K

m, n

such that this graph always contains a subgraph isomorphic to K

m vertices in the first class.

Since 1951 the Zarankiewicz function has been studied by many authors,

and a survey of the main results can be found in 11 . For the sake of our

purposes, we recall the following:

R m,n ? 2mn ? 1 ? 1.3

Ž. Ž .

Ž

Ž

.

.

a, a

with

m, n

??

L EMMA 1.

If

x ? 1

m

xa

m

?

? a ??

Ž

? n ? 1

Ž

,

..

ž/ž /

m

ž /

Ž .

a ? ax??.

The proof, based on the convexity of the binomial function, can

? ???

be found in Bollobas 4 or in Guy 11 .

Â

then Z

m, n

Proof.

2. UPPER BOUNDS FOR R

The next result establishes the elementary connection between the

bipartite Ramsey function and the Zarankiewicz function:

Ž .

?

?

L EMMA 2.

If Za ? a ?r then R m, n ? a, where z denotes the

m, n

least integer not less than z.

2

Ž.

? ?

r

??

Proof.

?

11 .

The case r ? 2 appears in 14 , and for generic r was studied in

?

THEOREM 3.

For e?ery r ? 2 and n ? 2,

R 1, n ? r n ? 1 ? 1,

Ž.

r

R 2, n ? r2n ? 1 ? r ? 1.

Ž.

r

Ž .

Item a is deduced from to the Dirichlet pigeonhole principle.

Ž.Ž

For b , consider x ? r n ? 1 , a ? r x ? 1 ? 1, and ? ? r ? 1 x ? 1 .

a

Ž. Ž .

Ž .

b

Ž.

Proof.

Ž ..Ž.Ž.

Page 4

RAMSEY PROBLEM

51

?

2

?

Ž .

a ? ax?

Because a ?r ? ax??, by Lemma 2, we shall prove that Z

?, which can be done using Lemma 1, by showing that its hypothesis

reduces to

2, n

2

2

xx ? 1 ? x r ? 1

.

? n ? 1

Ž

x ? 1

.

ŽŽ

.

.Ž.

r x ? 1 ? 1

Ž

r x ? 1 ? 2 .

Ž

4

. Ž .

Because n ? 1? x?r, dividing both sides by x and performing some

simple algebraic manipulations we obtain the equivalent inequality,

2? 2?r,5

Ž .

Ž .

Because r ? 2, the inequality 5 is true, and the result follows.

We remark that in the cases where r ? 2, the items a and b of the

preceding theorem give the same estimates as in 1a and 1b .

Ž

THEOREM 4 General Upper Bound for the Bipartite Ramsey Function .

For integers m ? 2 and n ? m,

Ž .Ž .

Ž.Ž.

.

R m, n ? 2mn ? 1 ? 2m?1? 1.

Ž.Ž.

Proof.

2n ? 1 2

Lemma 2, this estimate holds if Z

the right side of last inequality is equivalent to z ? z ? 1 , using

Lemma 1, it is sufficient to check that

ž /ž

mm

The cases m ? 2 and m ? 3 are well known. Defining z ?

m?2

, the result can be interpreted as R m, n ? 2z ? 1. By

Ž

2z ? 1 ? 2z ? 1 ?2 . Because

m, n

Ž.

Ž.

.

?Ž.2

Ž

?

2

.2

zz ? 12z ? 1

m

z

? z ? 1

Ž

? n ? 1

Ž

,

..

/ž/

or, equivalently,

2

z ? z ? 1

Ž

? n ? 1

Ž

z ? m

2z ? 1 ??? 2z ? m .

.Ž.

z ? 1 ??? z ? m

.

Ž

.Ž . ŽŽ.

6

.Ž .

Ž .

The rest of the proof consists in verifying the validity of 6 for every

m ? 4. First we tackle the case where m is odd.

Case 1.If m ? 2s ? 1, s ? 2. Because z ? z z ? 1 , rearranging the

Ž .

factors in 6 it is enough to show that

ž

z ? 1

i?1

2

Ž.

ss

2z ? 12z ? 2i

z ? i

2z ? 2i ? 1

z ? s ? i

2z ? m ? n ? 1 .

.

...7

ŽŽ. Ž .

ŁŁ

i?1

/ž/ž/

Page 5

CARNIELLI AND MONTE CARMELO

52

This arrangement of factors, which we note is slightly different from the

? ?

one given by Furedi 8 , makes possible to calculate the rates as follows:

È

Ž .

CLAIM 1.i Each factor in the first product is equal to 2

Ž .Ž . Ž.

ii2z ? 2i ? 1 ? z ? s ? i ? 2 1? 1? 2n ? 1 2 , 1? i ? s

Ž .Ž . Ž.

?

iii2z ? 1 ? z ? 1 ? 2 1? 1? 2n ? 1 2

.

1 2

Ž .

We shall prove item ii only: for s ? 2 and 1? i ? s,

?

Ž.

s?

?

Ž.

2s?1?

Ž

? 2 1? 1? 2n ?

s?

s ? i

ss

2s ? 1? 2 ? 1? 2 ?

.

s?1

2n ? 1 2

Ž.

Ž.

s?1

Multiplying the outside terms of the inequality by 2n ? 1 2

2s ? 1

z ? s ? i

Ž . Ž.

But 2z ? 2i ? 1 ? z ? s ? i ? 2? 2s ? 1 ? z ? s ? i , and this com-

Ž .

pletes the proof of item ii .

We now return to the proof of the theorem. Applying the result of

Ž .

Claim 1 in 7 , it is sufficient to note that

we obtain

1

?

.

s?1

2n ? 1 2

Ž

Ž.

. Ž.

s?1

2z ? m

n ? 1 2

.

1

? 1?

.8

Ž .

s

2s?1

2n ? 1 2

Ž.

Ž.

Ž

ss

Because 2z ? 2 n ? 1 2 ? 2 , the computation of the rate in the left side

of the inequality gives

s?1

112s ? 1

n ? 1 2

Ž.

1?? 1??

.9

Ž .

s

2s?1

2 n ? 1

Ž

Ž .

2n ? 1 2

.Ž.

Ž.

The proof of 7 can be reduced to two cases:

Case 1.1.

prove the validity of 9 . We start by computing the term between square

Ž.

brackets. Considering a ? 2n ? 1 2 , by the Newton binomial expansion

of this power series we obtain

Ýž

a

i?2

Because

s ? 1

? s ? 1 ,

ž/

i

s ? 3. From the previous considerations, it is enough to

Ž .

s

i

s?1

s ? 11

a

s ? 1

i

1??

.

/ž /

i

Ž.

Page 6

RAMSEY PROBLEM

53

we have

ii

1

a

s ? 1

a

s ? 1

i

?

ž/ž / ž/

1

2

s

Ž.

for 2? i ? s ? 1. As, by hypothesis, s ? 3, we get

thus,

/ž / ž

i?2

s ? 1 ?2 ?

and

2

s?1

i

s?1

Ýž

1

a

11

s ? 1

i

??????

/ž/

2 2n ? 1

Ž

ž

2 2n ? 1

Ž..

2

1

? s

.

/

2 2n ? 1

Ž.

Ž .

These results make 9 valid if

s ? 1

2

s

2s ? 1

n ? 1 2

Ž

2n ? 1

2s?1

.

Ž .Ž.

1???

s

4 2n ? 1

Ž.

holds. But this holds for n ? m ? 2s ? 1, because of the relations,

1

2

s ? 1

2s

1

4

s

1

4

2s ? 1

Ž

2n ? 1

s

.

Ž .Ž.

?

,

?

,

?

.

4. 2n ? 1

Ž

n ? 1 2

.

Ž.

Case 1.2.

improve 9 to

s ? 2. Using the best estimate in Claim 1iii , it is possible to

Ž .

2

11

a

1

a

5

1??

1?

1??

, 10

Ž.

ž / ž/

2 n ? 1

Ž

32. n ? 1

Ž..

Ž.

where a ? 4 2n ? 1 . Because

1511

16

12

3 2n ? 1

Ž

18

??

?

??

,

2 n ? 1

Ž

32 n ? 1

Ž

2 n ? 1

Ž

3a

....

it is sufficient to note, expanding the factor between brackets, that:

8? 3a ? 5?2a ? 2?a2? 1?2a3. Through elementary algebraic manipula-

tions, multiplying this inequality by a3we end up with a2? 12a ? 3? 0.

Ž.

Because a ? 4 2n ? 1 , it is easy to see that this last inequality is true for

all n ? m ? 5.

Page 7

CARNIELLI AND MONTE CARMELO

54

Case 2.

m ? 2s, s ? 2. By similar arguments it is enough to check that

2z ? 12z ? 2i

2z ? 2s ? n ? 1

.Ž.

Ł

ž

z ? 1

i?1

where the corresponding rates can be estimated by:

Ž .

CLAIM 2.i Each factor of the first product is equal to 2

Ž .Ž . Ž.

ii2z ? 2i ? 1 ? z ? s ? i ? 2 1? 1? 2n ? 1 2

? 1

Ž .Ž. Ž.

?

iii2z ? 1 ? z ? 1 ? 2 1? 1? 2n ? 1 2

.

1 2

ss?1

Ł

/ ž

i?1

2z ? 2i ? 1

z ? s ? i

, 11

ŽŽ.

/ ž/

z ? i

?

Ž.

s?1?

, 1? i ? s

Ž.

2s?2??

Ž

? 2 1? 1? 2n ?

s?1?

By means of a similar procedure, it suffices to verify that

s

112s

1?? 1??

. 12

Ž.

s?12s

2 n ? 1

Ž

2n ? 1 2

n ? 1 2

.

Ž.Ž.

Let us consider three cases:

Ž. Ž Ž

2 s ? 1 ? 1 ? n ?

.. Ž

Case 2.1.

2Žs?1.

1 2

For s ? 4,

Ž

? 2s? n ? 1 2

Case 2.2. For s ? 3. Observe that 5? 32. n ? 1 ? 6? 64. n ? 1 and

Ž.

use 10 .

12 holds. Observe that

and use the inequality 9 for s ? 1? 3.

ŽŽ

.

.

2s

Ž .

..ŽŽ..

Ž .

Case 2.3.

m ? 4, we proceed directly to 6 . Hence,

For s ? 2, because we cannot use 7 , since this is false for

Ž .

34

2

z ? z ? 1

Ž

z ? 4

z ? i ? n ? 1

.

2z ? i ,

.Ž.ŽŽ.Ž.

Ł

i?1

Ł

i?1

Ž.Ž .

where z ? 4 2n ? 1 . In the same way as we established 5 , a simple

analysis shows its validity for all z ? 4 2n ? 1 , n ? 4. This completes the

proof for all possible cases.

Ž.

We now discuss some immediate consequences of the previous theo-

rems. Given integers k ? 2 and 1? m ? n, let r m, n

smallest integer a such that every k-coloring of the edges of a complete

graph K

always contains a monochromatic copy of K

a

Ž.Ž? ?.

r m, n . Chung and Graham7 give some classes of upper bounds for

2

Ž.

the function r m, n ; in particular, for the case k ? 2,

Ž.Ž .Ž

ž

Ž.

denote the

k

Ž.

. Let r m, n ?

m, n

m

1?m

r m, n ? n ? 12? 2, 13a

.Ž.

m

m ? 1?2

n ? 1 2

r m, n ? n ? 1

Ž.

2?

. 13b

Ž.Ž.

m?2

/

Ž.

Page 8

RAMSEY PROBLEM

55

Ž.Ž.

From the elementary inequality r m, n ? 2.R m, n and the foregoing

theorem we derive

r m, n ? n ? 1 2m?1? 2m ? 2,

Ž.Ž.

Ž.

which sharpens asymptotically the bound

m ? n our estimates sharpen the bound due to Chvatal by 2

Ž.

r m, m ? 2.m.k

when k ? 2.

k

13a . In the diagonal case

Â

mŽ

? ?.

cf. 7 ,

m

3. LOWER BOUNDS FOR R

? ?

In the cases r ? 2 and m ? 2 or 3, 2 presents some classes of lower

Ž.

bounds for R m, n through Hadamard matrices. Also, 11 reports some

2

results on the polarized version of the bipartite Ramsey problem initially

?

investigated by Hales and Jewett 12 . In particular, certain results for

r ? 2 in connection with self-complementary designs are reported in 14 .

The main result of this section describes the asymptotic behavior of

Ž.

R 2,2 .

r

Before stating our next result, we introduce some notation and we make

Ž .

some remarks. Let GF q denote the finite field containing q elements,

and let ?

be the ring of integers modulo q. It is well known that there

q

exist q ? 1 mutually orthogonal Latin squares of order q with entries in

Ž . Ž

? ???.

GF q

see 3 or 15 . Any fixed bijection from GF q into ? transforms

such matrices into q ? 1 Latin squares, say A , A ,..., A

in ? . Let A

be a matrix of order q defined by A a, b ? b, 0? a,

q

0

b ? q ? 1.

THEOREM 5.

For q a prime or prime power,

??

?

??

Ž .

q

with entries

12

q?1

Ž.

0

R

2,2 ? q2.

.Ž

q

Proof.

q-coloration of K

will be to define a function F: V ? V ? 0,1,..., q ? 1 , where V ? ? ?

?

represents both classes of vertices in the bipartite graph, F

q

represents the collection of edges having color j such that F

edges of any subgraph isomorphic to K

function F as follows: for each edge a, b of V and for each color j, put

ŽŽ . Ž..

F a, b , c, d

? j if and only if d ? A a, b ? j, that is, iff

?

0

In order to prove the theorem it is sufficient to exhibit a

2 which avoids a monochromatic K

q , q

?

2

; our strategy

2, 2

4

q

?1? 4

j

?1? 4

j has no

, 0? j ? q ? 1. Define the

2, 2

.Ž

Ž.

c

c, d ?

.

0, Aa, b ? j , 1, A a, b ? j ,...,

.

. Ž

1

Ž

ŽŽŽ.

Ž.

q ? 1, Aa, b ? j

.

. 14

ŽŽ.

4

.

q?1

Page 9

CARNIELLI AND MONTE CARMELO

56

By the construction, F is a q-coloring of V ? V. Indeed, for each given

Ž.

vertex a, b , we can observe that V coincides with the disjoint union of the

Ž.

sets defined in 14 , indexed by the colors j, 0? j ? q ? 1. The final steps

of the proof consist in verifying that there can be no set of the form

?4?4

u,? ? x, y , u ??, x ? y and color j such that u,? ? x, y ? F

We check only the color j ? 0, because the symmetry of F guarantees no

Ž.

loss of generality. For each a, b in V, denote by T

obtained by the q second coordinates of the pairs defined in 14

.

fixed ordering of such pairs . By the construction, the set C ? T : u ? V

Ž.q

is a d-Latin code in ?

, or equivalently, a maximum distance separator

q

.

code , where d ? q ? 1 represents the Hamming distance of the code C

Ž

for the definitions and results see 5 and 15 . Hence the Hamming

distance between two distinct words in C must be greater or equal than

q ? 1. If this is not the case, then there exists u,? ? x, y such that

?4?4

Ž.

u,? ? x, y ? F

0 . By 14 , these four vertices would have to satisfy

?4?4

?1? 4

j .

Ž

for a

.q

the vector in ?

Ža, b.

q

Ž . Ž

?4

u

? ???.

?4?4

?1? 4

x ? A

2

u ,

Ž .

1

Ž .

1

y ? A

2

u ,

Ž .

1

Ž .

1

xy

x ? A

2

? ,

y ? A

2

? ,

xy

Ž.Ž.

where x ? x , x

and T in the code C coincide in the coordinates x and y whose values,

?

respectively, are x and y . In this case, the Hamming distance between

22

these two words is less than q ? 1, which contradicts the fact stated

previously. This completes the proof.

and y ? y , y . Observing that x ? y , the words T

121211

u

11

The next result is inspired by a limit of the Zarankiewicz function due to

??

Mors 13 .

È

We ha?e

COROLLARY 6.

lim R

n??

2,2 .n?2? 1.

.Ž

n

Proof.

sufficiently large n such that

We must prove that, for a given 0? ? ? 1, there exists a

R

2,2

n

Ž.

n

1? ? ?? 1? ?.

2

Ž .

Ž .

Indeed, the inequality on the right side follows from Theorem 3 b ,

Ž.Ž .

because R 2,2 ? n ? O n . On the other hand, consider ? ? ? ?

n

Ž.2

satisfying 1? ? ? 1? ?

. For such ?, and for sufficiently large n, there

exists a prime number p such that 1? ? n ? p ? n we note that an

n

?

analogous argument was used in 13 . Now, the monotonicity of R, the

2

Ž

?.

.Ž

n

Page 10

RAMSEY PROBLEM

57

previous theorem and the last inequality imply

R

2,2

n

2

n

2

R

2,2

n

p

n

Ž.

Ž.

p

n

2

n

??? 1? ?

Ž

? 1? ?,

.

22

and so the desired result follows.

Ž .

It is appropriate to observe that, simultaneously applying Theorems 3 b

and 5, we can obtain the relation:

COROLLARY 7.

For e?ery prime or power prime q,

Ž.

q

q2? R

2,2 ? q q ? 1 ? 1.

Ž.

This simple relation allows us to estimate bounds very close to the exact

values. So, for example, one obtains R 2,2 ? 5, R 2,2 ? 10 or 11,

Ž.

17? R 2,2 ? 19. For certain subclasses, however, we can obtain sharper

4

lower bounds, as shown in the following text:

Ž.Ž.

3

THEOREM 8.

For q prime,

R

2,2 ? q q ? 1 .

.ŽŽ.

q?1

Proof.

? V ?

avoids the appearance of a monochromatic K

of the coloring function F defined in the preceding text. For every a, b in

Ž

V and 0? j ? q ? 1, denote by B a, b ? j the set defined in 14 by Latin

Ž.

squares A x, y ? i.x ? y, 0? i ? q ? 1. Let B a, b ? j be the block

i

Ž.

i.e., a set of cardinality q

defined in the following way: replace the

Ž.

element of B a, b ? j whose first coordinate is a ? j by the element

Ž.

.Ž

q, a ? j , that is: B a, b ? j ? B a, b ? j ? a ? j, a ? j .a ? b ? j

?Ž.4Ž.

?

q, a ? j . For a fixed

a, b , varying j, it is easy to see that the

elements which have been substituted are

?

Ž

The proof is based on the previous theorem: let W ? ?

Ž? 4.

q ? ? , we construct a coloring f: W ? W ? 0,1,..., q which

q

? ?

q?1

q

?4

by suitable modifications

2, 2

Ž.

.Ž.

?Ž.

Ž.

?Ž. ?ŽŽ..4

a ? j, a2? b ? a ? 1 .j , 0? j ? q ? 1 ,

Ž.

4

.

Ž.

which, by simple inspection, coincide with the elements of B a ? 1, b ? a .

Now we are in position to exhibit the desired function f. Because

? 4 ŽŽ. Ž

W ? V ? q ? ? , define f a, b , c, d

q

?

½

?

c, d ?

??

..

? j if and only if

5

a ? q

?

B a, b ? j

Ž

B a, b ? j

Ž

4

b ? j ? ? ,

0? j ? q ? 1

j ? q

0? j ? q,

.

.

,

0? a ? q ? 1,

Ž.

q

Page 11

CARNIELLI AND MONTE CARMELO

58

where ? represents addition in ?

serves the properties,

. Thus the construction of f pre-

q?1

Ž

Ž

Ž

.

.

.

?? 4

k ? ? ? j ? ? ? 0 for k ? j.

q

?? 4Ž

k ? ? ? B a, b ? 1

q

? Ž . Ž .?

B u ? B ? ? 1 for u ?? of V.

? 4

?

p1

p2

p3

q

.?

Moreover, the construction satisfies:

Ž.

?

?Ž .

B u ? B ? ? 1 for u ?? of V.

?Ž .?

p4

Ž.

The first two are immediate, while

Ž.

theorem. For p4 , weargue asfollows: supposethat x, y ? B u ? B ? ;

?

in this situation, the possibilities

x, y ? V and

Ž.Ž

excluded, due to properties p3 and p1 , respectively. There only remains

? 4

the case x ? V and y ? q ? ? . By construction, y occurs in q blocks,

q

?Ž .Ž .

say B 1 , B 2 ,..., B q . But B 1 , B 2 ,..., B q partitions V and con-

Ž

tains the set B 1 ? B 2 ????? B q ? q ? ? . Hence among these

q blocks one cannot find two distinct blocks containing x. The four

previous properties guarantee that there exists no monochromatic u,? ?

?4

x, y in f.

p3 was proved in the previous

?4

4?4

x, y ? q ? ?

.

?Ž .

? 4

?Ž .

are

q

?Ž .

?Ž . Ž .

?Ž .. ? 4

Ž .

?Ž .

?Ž .

q

?4

ACKNOWLEDGMENT

We thank Y oshiharu Kohayakawa from the IME-University of Sao Paulo for pointing out

some of the references.

Ä

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Ž.

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Â

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È

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Ž.

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