A Canonical Ramsey Theorem

Nancy Eaton and Vojtkh Rod1 Department of Mathematics, Emory University, Atlanta GA 30322

ABSTRACT

Say a graph H selects a graph G if given any coloring of H , there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H . Denote by s(G) the minimum order of a graph that selects G and set s(n) = max{s(G) : ICl = n } . Upper and lower bounds a re given for this function. Also, consider the Folkman function f , (n) = max{min{IV(H)I : H+(G) j } : IV(G)l = n } , where H+(G),! indicates that H is vertex Ramsey to G, that is, any vertcx coloring of H with r colors admits a monochromatic induced copy of G. The method used provides a better upper bound for this function than was previously known. As a tool, we establish a theorem for projective planes. 0 1992 John Wiley & Sons. Inc.

1. DESCRIPTION OF RESULTS

In this article we consider an analogue of Ramsey numbers for vertex colorings of graphs and show that the random graph approach yields the graphs that satisfy some Ramsey properties of interest. If G and H are graphs and r a positive integer, we write H+ (G): to mean that any coloring of the vertices of H by r colors yields a monochromatic induced copy of G. In [8] J . Folkman proved that for every integer r and graph G, there exists a graph H so that H-;. (G); holds.

Here we investigate the numerical aspect of the problem. For a graph G and a integer r , let f ,(G) denote the Folkman function, i.e., the minimum order of a graph H such that H-. (G): and let f , (n ) = max{ f , (G) : IV(G)l= n } . Thc ques- tion is, how does f,(n) behave.

We will also consider the following extension of the problem. In [7], Erdos and Rado proved the Canonization theorem which extends the Ramsey theorem (see [15]) to the consideration when the number of colors used is not fixed. Note that the vertex version of this is the following form of the pigeon hole principle: For every n , there exists an m such that if X is a set, 1x1 = rn and X = X, U X , U

Random Structures and Algorithms, Vol. 3, No. 4 (1992) 0 1992 John Wiley & Sons, Inc. CCC 1042-9832/92/040427-18$04.00

428 EATON AND RODL

* * U X , is a coloring of its elements then there exists Y C X , I YI = n with either (i) Y c ~ for some i E { l , 2 , . . . , r } or (ii) p n x J s i , V i E { 1 , 2 , . . . , r } . Clearly rn 2 (a - I)* + I .

Similarly to the way Erdos and Rado extent and Ramsey Theorem, Folkman’s result has been extended by J. NeSetiil and V. Rodl. In [13] a version of Folkman’s theorem was established where the number of colors is unrestricted by showing that for every graph G, there exists a graph H such that any coloring of the vertices of H (with an arbitrary number of colors) yields an induced copy of G which is either monochromatic or totally multicolored, i.e., each color class contains at most one vertex. We call such a graph H a selective graph for G and say H selects G. Note that these graphs were further studied in [6] and their properties were used in [14] in connection with a problem in topology.

Let s(G) denote the minimum order of a graph that selects G and set

s(n) = max{s(G) : I v ( c ) ~ = n}

The f ,(G) and s(G) give the Ramsey numbers of G if we color vertices by r or an arbitrary number of colors, respectively. While f,(n) and s(n) describe the growth of these functions in the worst case over all graphs G on n vertices.

The aim of this article is to give an estimate of the function s(n) . We prove the following:

Theorem 1. There exist constants c1 and c2 such that

3 4 c ln ~ s ( n ) ~ c , n logn

holds for all n.

As a consequence of our proof, we obtain:

Corollary 2. There exists an absolute constant c such that for any fixed r ,

This is an improvement over the earlier result of J. Brown and V. Rodl (see (31) who established that

where c , and c2 are absolute constants.

Note that the relation H+ (G): has been investigated from a different point of view in [l], [2], [5] , and [ll] where the concept of G-chromatic number has been investigated. Let G and H be graphs and r an integer. The G-chromatic number of H , x c ( H ) , is the least r such that for every coloring V ( H ) = Vl U V, U U V,-, , there exists an i E {1,2, . . . , r - 1} such that the subgraph of H induced by the vertices, V , , contains an induced subgraph isomorphic to G. From this point of view, f ,(G) denotes the minimum order of a graph H with ,y , (H) > r . Note

CANONICAL RAMSEY THEOREM 429

that when G is tan edge, this concept coincides with the ordinary chromatic number of a graph.

2. A RESULT FOR PROJECTIVE PLANES

Here we present a result on the incidence of a set of points in a projective plane to a set of lines, giving bounds on the number of 1’s in a submatrix of the incidence matrix of the plane. This result will be used in the proof of the main theorem. It extends Lemma 2.3 from [3] . Note that a similar approach was used for Hadamard Matrices in [9]. For an exposition on projective planes, see [12].

Theorem 3. Let 9 = (V, 2) be a projective plane of order p and let N = p 2 + p + 1 be the number of points in the plane. Suppose X C V, 1 XI = a N , and 9 C 2, 191 = P N , then

where o( 1 ) is a function that goes to zero as p goes to in$nity.

Proof. B = ( A + l ) A - I where J is the all 1’s matrix.

i # j , we infer that

Let A be the incidence matrix for the plane, (V, 2). Let A = p + and

To see that the rows of B are orthogonal, let bi be the ith row of B . Then for

( b , , b , ) = A 2 + ( - 1 ) h 2 p + ( - 1 ) 2 ( p 2 + p + 1 - 2 p - 1 ) = 0

Let C be the submatrix of B formed by the rows in X and columns in Y . For ease of notation, assume C is the upper left-hand corner of B . Let c, be the ith row of C and let ( <,, &, . . . , S O N ) = c, + c2 + . . . + ceN. To obtain the estimate, we first consider,

Due to the orthoganality of the rows of B , we have,

B N

= a ~ ( ~ 2 ( p + I) + ( - q 2 p 2 )

and since N = p 2 + p + 1 ,

430

On the other hand, by the Cauchy-Schwarz inequality,

EATON AND ReDL

(1)

and

From (1) and (2), we obtain,

which gives the result. w

3. PROOF OF THEOREM 1

Proof. First we show the lower bound. Note that if H selects G, where G is a graph with n vertices, then also H+ (G);-, holds, since if colored by n - 1 colors, H cannot contain a multicolored copy of G. It was, however, shown in [3] that f,(G) 2 ( r - l )n2/4 + n for G = Kn,’ + D n i Z . Hence we get s(n) 2J,-l(n) 2

cn3 where c is an absolute constant. Next, we establish the upper bound. To do so, we must show that for any graph

G on n vertices, there is a graph H on no more than Cn4 log n vertices which selects G.

We define the variables, z , x, and a! as follows,

( 3 ) 1

2n z = [20nlogn], x = (Slogn] and a! = - .

Choose p to be a prime satisfying,

nz - - 1 4 p 5 2( y - 1) . X

So we have,

CANONICAL RAMSEY THEOKEM 431

Set

N = p 2 + p + 1 a n d t = L"'pn+ l)1.

As a consequence of these definitions we have the following relationships that will be used throughout the proof:

We will consider the incidence structure (W, A ) defined as follows:

Definition 1. For a fixed plane, 9 = (V, 2) of order p consider its x copies,

For v E V, let

be a set of copies of u , called the expansion of u . Then we set

Let V ( G ) = { u , , u Z , . . . , u , ~ } and E ( G ) be the vertices and edges of G, respectively, and let T, , T2, . , . , T, be disjoint sets, where 1 TL I = t for every i E [ n ] = { 1 , 2 , . . . , n}. We consider the auxiliary n-partite graph G(tj defined by:

and

E(G(t)) = { { U, v} : u E T, , u E T, , { U , , u,} E E ( G ) , i # j }

We now describe a class of graphs created on the vertex set W. For a given M E A, let L E Y be such that M = U W,. Consider a one-to-one mapping, + M : V(G(t))+ M . Note that IV(G(t))l= nt s x ( p + 1) = IMI and so the maps exist. For each such mapping define the graph H ( I ) ~ ) by,

and

EATON AND R6DL 432

In other words, the edge set of H(GM) consists of those I + % ~ images of edges of G ( t ) which are not contained in W, for any u E L. Hence, for all u E L , W, is an independent set of points in H(I,!J~).

Let [.Ik indicate the falling factorial of length k , [nIk = n(n - 1) * . . ( n - k + 1).

Definition 2. Consider the space of all

N-tuples, A } assign a graph Hw given by,

= { CCI, : M E A } , each being equally likely. To each !P = { GM : M E

(Since t l u E V , W, is an independent set, this graph is well defined.)

happen that Hw, = Hq2 f o r distinct H E X are equally likely.)

Let X be the space of all graphs Hw obtained in this way. (Note that it may and q2, and as a consequence not all graphs

We will use a probabilistic method to show that at least one graph H E X will satisfy our requirements. The argument is broken up into two cases determined by the number of colors used to color the graph H . In each case it is shown that the probability that a random H E X satisfies the theorem is bigger than - which implies that there is some H E 2t satisfying the theorem.

1 2

Let H E X and suppose the vertices of H are colored with r colors.

Case 1. The number of color classes, r , is less than or equal to 2n.

In view of (3), l l r 1 a and thus one of the color classes has cardinality at least a1 Wl = axN. Let E be the event that there exists a subset X C W of size axN that does not contain an induced copy of G. We will show that

1 2 Prob(E) < - .

For a given line, M E Ad and set X , let A M be the event that the graph H ( G M ) does not contain an induced copy of G. Then since for M # M', the events A M and AM, are independent, we see that

Prob(E) 5 (*) n Prob(A,). axN M€.&

Fix X C W and M E A. Let L be such that M = U W,. Let { u l , u 2 , . . . , u k } C L be the set of all u E L with W, n X # 0 and for each i E [ k ] , set C, = X fl W,,. Thus clearly U :zl C, constitutes a partition of X n M . For each i E [n] and one-to-one map GM : U :=, T, + M, let

CANONICAL KAMSEY THEOREM 433

Then I, indicates which classes among C,, C,, . . . , C , from a nonempty intersec- tion with $,(TI). See Figure 1.

. . . , Zn($,)} has no system of distinct representatives. Clearly, B, implies A,. We will find an upper bound on the probability of B, and as Prob (B , ) 2 Prob(A,), this will yield an upper bound on the probability of A,.

Thus, Prob(B,) is equal to the number of one-to-one maps $,,,, : V(G(t))+ M that produce no S.D.R. In 9(+,) divided by the total number of one-to-one maps. By Hall's Theorem (see [lo]), for a given map $,, there is no S.D.R. of ,a($,+,) if there exists J C [ n ] , IJl = j , such that 1 U l t , Z,($,)I < j , which implies that there is an index set J ' C [ k ] , IJ'I = j - 1 such that,

Let B, bc the event that the set = {f1($,,,,),

$,( u l E J T ) c UIE,' c, lJ ( M \ ( X n W ) . (9)

See Figure 2 which illustrates this situation for j = 4.

M

I

Fig. 2. J' = { i l , i,, i3}

EATON AND RODL 434

Thus in order to estimate Prob(B,) we need to find the probability that a random $M satisfies (9). We will find a small upper bound for Prob(B,) for those M which we call good lines (and use the trivial fact that Prob(B,) 5 1 for all other M ) .

9 Definition 3. The "line," M E Jd is a good line i f { IX n MI 2 ~ a ( p + 1)x.

That is to say, the size of the intersection is almost the average of { I X n

We will use the following two claims: MI : M E A } which clearly is equal to a( p + 1)x.

Claim 1.1. Let M be a good line, then for large enough n

1 n

Prob(B,) 5 - .

Claim 1.2. There are at least N / 2 distinct good lines M E A.

Assuming Claims 1.1 and 1.2, the rest of the proof is easy:

By Claims 1.1 and 1.2, we can define 3' C Ad such that 191 = N / 2 and V L E 9, Prob(A,) c l l n , so from (8),

N / 2 N i 2

Prob(E) 5 ( xN )( :) n Prob(A,) 5 ( zx%)(i) (YXN M@?4

-N i2 log II 5 (z)"'" e

axN - axN log (Y - - " 1 log n . 2

So, Prob(E) < 1 /2 whenever

N 2 a x N - axNlog a - - log n < -1

N 2 x ( a N - aNlog a)< - log n - 1

and as by (3), a = 1 /2n we infer that

n(N log n - 2) (1 + log(2n))N X <

Note that the RHS of (10) is asymptotic to n while x = 1.5 log nJ . Thus there exists n, such that (10) is true for n 2 no.

Proof of Claim 1.1. For the good line M let rn = I U i e J . Ci U ( M \ ( X n M)) I .

Since lC,l I x for each i E J ' , ]MI = ( p + 1)x and IX n MI 2 (Y -( p + l)x, we have by (9),

9 10

CANONICAL RAMSEY THEOREM 435

9 10 m 5 ( j - 1)x + ( p + 1)x - a - ( p + 1)x ,

l )x+ 1- ff - ( p + l ) x . [ ;01

For a fixed subset, J C [n] of size j and subset, J ’ C [ k ] of size j - 1, the number of one-to-one maps $M : V(G(t))+ M that satisfy (9) over the total number of one-to-one maps (in other words, the probability that a fixed jt element set U ,El T, is mapped into a fixed m-element set U L E J j C, U (M\(X f l M ) ) is given by:

( j - 1)x + [l - a & ] ( p + 1)x 1‘ 4 ( P + 1)x 1 . Now, letting J C [n] and J ‘ C [ k ] range over all sets with IJI = IJ’( + 1 = j ,

where l I j 5 m i n { k + 1 , n } , wesee tha t

As k s p + 1 and due to (7) we get that

On the other hand, by (3), (6), (7 ) , and the fact that j - 1 < n we get that

j - 1 + [I - G a ] ( p + 1)

5 np1’’’ .

Considering (12), (13), and (14) we infer that there exists n 1 such that

1 Prob(B,) 5 c (80n-’”’)’ < -

n I

for n r n l .

436 EATON AND RODL

Proof of Claim 1.2. Suppose there are less than N / 2 lines, M , with the property 9

that IXn MI 2 - a x ( p + 1) and hence there is a set 9 C A, 191 2 N l 2 , such 10 9

that IX n MI < -ax( p + 1) for each M E A. Thus, 10

Note that by definition 1 the underlying set, W is a union of the sets V,, V,, . . . , V,. we will set 9, = { M n V , : M E 9}. Clearly, 1911 2 N / 2 and 911 C

For each i E [XI, let a , N be the number of points in X from the ith plane. In =%.

other words, a,N = [ X U V,l, giving the relationship.

a x N = (a , + cr2 + . . . + a, )N.

Then

and thus by theorem ( 3 ) ,

wherc o(1) is a function that goes to zero as p goes to infinity On the other hand, by the Cauchy-Schwartz inequality,

I12

i = l i = 1

Then from (15), (16), (17), and (18),

Giving,

CANONICAL RAMSEY THEOREM 437

Due to (7) and the choice of a we have

and thus, because V% - p we infer that

r

On the other hand,

N ( P + l ) ( P + * + l ) - l

and thus the RHS of (19) tends to zero as p increases which implies that n increases. Thus, there exists n2 such that n 2 nz implies that our assumption on the cardinality of the set 3 leads to a contradiction, proving the claim.

Case 2. The number of color classes, r , is larger than 2n.

We can assume that the size of each color class is less than x N i 2 n , since if not there would be a set X as in Case 1. In this case, we are going to prove that a multicolored induced copy of G must exist in more than 1/2 of the graphs H E 2%’.

It is more convenient to work with classes of “uniform sizes.” This can be obtained by combining pairs of color classes of size less than xNi4n until we get k classes, U , , U,, . . . , U,, with

so that

for every i E [ k ] . Note that the size of each “line” M E A is x( p + 1) = xV%( 1 + o( l)), where

o( l )+ 0 as p - f ~0 and thus the average size of M n U, , i E [ k ] is at least (1 + o(l))xV%/4n.

Definition 4. Define a full line to be a line, M E A such that V i E [ k ] , 1 U, n MI 2 x f l l 4 0 n (i.e., U, f l M attains at least one tenth of its average size).

Claim 2.1. When n is large enough, there are N i2 full lines in A.

Proof of Claim 2.1. Fix i E [ k ] , let ”Yi C A be the set of all “lines” that intersect U, in less than x f l l 4 0 n points. Set PEN = 1 % , I , and aixN = 1 U,l. Note that due to (21),

114n 5 ai 5 1 /2n

holds.

438 EATON AND RODL

Again, consider the structure (W, &) to be the x projective planes PI, P27 . . . Yx as in Definition 1. For each j E [XI, let 941,, = { L E 2, : M f l V , = L , M E ?I,}, i.e., 9Iz,] is formed by “traces” of “y, in the jth plane, 9,. This gives the relationship 191,Jl = = P , N . Let C Y , , ~ N = lU, n ”;. As clearly, lU,( =

X

I -

Moreover, X

and so by Theorem 3,

where o( 1)- 0 as N 4 a (or equivalently p -+ a). And as

we have

Combining this and (18) we get

Solving for fl we get,

40n(l + o(1))

(40-n - q v ” ~ * and since 40-n - 1 /* is increasing in a, and due to (22) 114n 5 a,, we havc,

400n (1 + o(1)) > Pi ‘ 81 fi

Let T be the total number of lines that are not full. Then we see that k

1=1

<- 400nk f i ( 1 + o(1)). 8

CANONICAL RAMSEY THEOREM 439

And so by ( 5 ) , (7), and (20) there exists an n3 such that,

1600 82

40 81

T < - n2VB(1 + o(1))

5 - ( p + l ) rn( l+ o(1))

1 < - N 2

for n 2 n 3 , giving the claim. For a full h e M , we will now estimate the probability of the event that there is

no totally multicolored induced copy of G in M . Let L €2 be such that U W, = M . We wish to start by finding a subset M' C M that intersects each W, in at most one point [see Fig. 3(A)], and moreover is uniform in the sense that each color class Ui intersects M' in the same number of points [see Fig. 3(B)]. For this we will use linear programming techniques.

Claim 2.2. there exists M' C M sutisfying

Let M be a full line and L E 2 be such that U W, = M . Then

and

IW, n M ' ( 5 1 for every u E L .

Proof of CZazrn 2.2. Let { W,, W,, . , . , W,+, } = { Wu : u E L } Consider the follow- ing bipartite multidigraph, D, with vertices, V , = { U l , U 2 , . . . , U,} and V, =

{ W,, W,, . . . , W,,,} and which contains lU, fl W,] arcs from U, to W, for all i E [ k ] and j E [ p + 13. From this description, we have that the in-degree of each vertex in V . is x and for all i E [ k ] , the out-degree of U, is 1 U,l. In this way, we obtain a one-to-one correspondence 4 for the set of these arcs and vertices of M .

Since M is full, we know that the out-degree of each vertex, U, of V, is given by

Fig. 3. (A) M' C M and IW" n M ' ( 5 1 . ( 6 ) (U, f l M'l = u, tfi.

440 EATON AND RODL

We will construct a network as follows. Consider a new vertex, g, and let the set of sources by V, U { g} and let V, be the set of sinks. We have all of the arcs described above for the digraph, and also let there be x arcs leading from g to each vertex in V,. For all i E [ k ] , let LIU,I/x] be the production at source U,, and let the demand be 1 at each sink in W,. The sum productions at the vertices U,, l s i s k i s

where 0 5 a I k and so let the production at g be a. With this assignment of integer valued productions and demands to the sources

and sinks in the network, we have the following feasible solution (see [4, Part 1111) :

(1) For each i , let any x 11 Uil /xl arcs lcaving U, be assigned a flow of 1 /x, and

(2) For each arc from U, with flow 0, if this arc leads to Wj, assign a flow of 1 lx

(3) All unlabeled arcs are assigned a flow of 0.

the rest a flow of 0.

to an arc from g to W,.

With this assignment, the productions are met as well as the demands. By the integrality Theorem (see [4, Part 111, p. 3271) there is an integer valued

feasible solution. Because the sinks, W,, W,, . . . , W,+ I , each have a demand of 1 and because of the structure of the network, this integer solution must correspond to all arcs in the network having a flow of either 0 or 1. Thus, each sink vertex in V, has exactly one arc with a flow of 1 entering it and each source vertex, U,, has exactly 11 U,l /XI arcs with a flow of 1 leaving it and the source vertex g and a arcs labeled 1 leaving it. Thus the set of all arcs with a flow of 1 , together with the vertex sets, V, and V,, form a forest which is a union of stars. Discarding the star which contains vertex g, and taking the 4-images of the remaining arcs yields the

disjoint sets, U ; , U ; , . . . , ULwith U : C U , , ILI:I=L~-~[IU,l/.x] and I yn u,=, U:l 51. So for M ' = U U : , the claim follows.

For a fixed coloring, and a fixed line, M E A, we define A, to be the event that no totally multicolored induced copy of C exists in (see Definition 2). The claim below gives an upper bound on Prob (AM).

V5-v k

Claim 2.3. For a full line M , and n large enough,

4n2 Prob(A,) 5 -p .

Proof of Claim 2.3. As we did in Case 1, we again make use of Philip Hall's Theorem, to find an upper bound for Prob(A,). Let U ; , U ; , . . . , U ; and M' = U U I be as constructed in the proof of Claim 2.2 with for all i E [ k ] ,

CANONICAL RAMSEY THEOREM 441

For each i E [n] and : n Ti+ M , let

Then I t ( + , ) indicates which classes among U ; , U ; , . . . , U ; form a nonempty intersection with +,(Tr) .

= {Zl(+,), 12(&,,), . . . , In(+,)} has no system of distinct representatives. Since B , implies A , we will find an upper bound on the probability of B , and similarly to Casc 1, this will yield an upper bound on the probability of A,.

Z,I < IJI. In other words, this means that there exist sets J C [n] and J ' C [ k ] , IJ'I = IJI - 1 such that.

Let B , be the event that the set

By Hall's theorem, B, occurs if there exists J C [n] such that I U

+( u r E J Tr 1 c u I t T ' u: u ( M \ M ' ) . (24)

For fixed sets J and J ' , let B, ,,,, be the event that $,,,, satisfies (24).

As 1 UIEJTIl = ti

IUrE,' u:u(M\M')1=IMI-lM'I+IU,,,, u:l

and by (23)

= x ( p + 1 ) - ku + ( j - 1 ) u .

Similarly to (11) we have that

[x(p + 1) - ku + ( i - 1 1 4 [x ( P + l)LJ

x ( p + 3) - ku + ( j - 1)u

Prob(B,,,yJ =

4 x ( p + 1 )

p - 40n 40 n

40n2 - 40n - 1 40n

>-

2

> n - 2 .

442

BY (417

EATON AND RODL

we get in view of (25) and the above inequalities that

Thus there exists n4 for which n > n4 yields

Prob(B,) < 4n’cxp( - ,

giving the claim. Let E be the event that there exists a coloring, such that for all “lines” M , A ,

happens, that is, no line contains a totally multicolored induced copy of G. To estimate the probability that for a randomly chosen H E X, for all colorings (with color classes smaller than x N / 2 n ) there exists a “line” M with a totally multico- lored copy of G, we will given an upper bound on the probability of the event E.

As the events A,, M E A are mutually independent this upper bound is given by,

Prob(E)% 2 n Prob(A,) all colorings M E &

s (xNjXN fl Prob(A,) n Prob(A,)

I (xN)”N(Prob(A,j)”2

M full A4 not full

CANONICAL RAMSEY THEOREM 443

To finish this case, we need to see that this last inequality is less than 112. By (31, ( 5 ) , and (7) we have that

N I 6400n4 and xN 9 3200n4 log n

so 2 N i 2

Prob(E) 4 xN“”( $1

( 3200n4 log n)log “4n2 eni3

= o(1).

Thus there exists ns such that for n 2 n 5 , Prob(E) < a . Note that the proof of Corollary 1.2 is obtained from Case 1 by replacing a

with l / r and z with 10r log n. Then if a graph G is colored by Y colors, we see that there exists a graph H that is vertex Ramsey to G that has x( p 2 + p + 1) vertices. By (3) and (4) we have

(2nzj2 5-

X

= 800r2n2 log n .

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Received February 3, 1992