On a classification of denumerable order types and

FUNDAMENTA MATHEMATICAE LI (1962)

On a classification of denumerable order types and an application to the partition calculus by

P . Erdős and A. Hajnal (Budapest) 1 . Introduction . In this paper ti-e are going to give a classification of denumerable order types, namely we are going to prove that every order type of a denumerable set which does not contain a dense subset can be built up from the order types 0, 1 by a transfinite induction process taking at every step the so-called (o-sic-m and *-stem of order types previously defined . Thus every order type O of such kind can have an ordinal number o (O) less than o) 17 called the rank o f O, associated with it-and several properties of denumerable order types can be verified by carrying out a transfinite induction on Q(0) ( 1 ) . As an application of the above-mentioned result a problem in the partition calculus for sets will be solved . Finally we are going to state some unsolved problems concerning non denumerable types ( 2 ) . 2 . Notations . Definitions . We are going to use the usual notations of set theory and we list only those where there is a danger of misunderstanding . Capital Roman letters denote sets, x, y, . . ., a, b, . . . denote elements of sets, a, (3, denote ordinal numbers, O, 99, 0 denote order types, n, k, i denote non-negative integers . -No distinction will be made between finite cardinal numbers and ordinal numbers . 21 will denote the type of rational numbers ordered according to magn it ude . X, ~ denote the cardinal number of X and T' respectively . If S is a set ordered by a relation R, then for an arbitrary pair x, y E S "x is less than y" will be denoted by x < y (R) and the order type of X will be denoted by X (R) . If there is no danger of misunderstanding (R) will be omitted . (1 ) This classification seems to be described somewhere in the literature ; Therefore it seems worthwhile to rive (3) For another application of the

so simple and natural that probably it is already however, the authors have been unable to find it . the proofs in detail . classification see (1] .



118

Y . Erdös and A . Hajnal

If an ordered set of type 0, contains a subset of type 0 2 we briefly write 02 - 0 1 . If S is a set ordered by the relation R and A, B C S then A B(R) denotes that a< b (R) for every pair a e A, b E B . DEFINITION 2 .1 . Let Z be a set 2 = T (T) and let O x bee defined for every x E Z . We define O = Z Ox as usual in the following way . XCZ

Let 8x be a system of disjoint sets ordered by the relations Rx such that AS;T = Ox (Rx ) for every x E Z . Then O is the type of the set S = U Sx XEZ

ordered by the following relation R . Let a, b E S, a E Sx , b E 5,,, a < b(R) if and only if either x < y(T) or x = ?f and a < b (Rx ) . It is well known that O depends only on the ordered set Z and on the function O x . O will be briefly termed a sure of type 9) of the Ox 's . If T = w or (p = a)* we may denote the Ox 's by O,, and we can speak of the co-sunz, or w*-sum of the sequence which will be denoted by

0, -; . . . + 0" - + 0.,, =- . . . = Oa ,

respectively .

R e m a r k s . 1 . It O, = y, for every x E w, then 0 depends only on 99 and r, , and %will be denoted by ~, as usual . 2 . Note that some of the Ox 's may be equal to 0, and thus, e .g . co < Oo + . . . + 0,z + . . . does not follow from Definition 2 .1 . Now we are going to redefine the partition symbol defined in [2] in the special cases needed for our purpose . Let [X]"n denote the set IY : T C X and I = na} DEFINITION 2 .2 . 0 1 - (0,", O3 ) 2 indicates that the following statement true . Whenever S is in ordered set, S = 0, and [S] 2 = I 1 1, is a partition of the set [S] 2 , then either there exists a set S' C S, S' = 0 2 such that [S'] 2 C 11 or there exists an S" C S, S" = 0 3 such that [S " ] 2 C I2 . O i v (0" 03 )' denotes the negation of the above statement. If )ra 17 na27 rn. 3 are cardinal numbers, then the symbol 7n 1 -->-(rn.27 m 3 )2 has a similar self-explanatory meaning. However, in this paper we are going to deal with the case when types and cardinals may appear in the same symbol . i

DEFINITION 2 .3 . Let O, 0, be types and let m be a cardinal number . O-*(0 17 in)" indicates the following statement . Whenever S is an ordered set, S = O and [S] 2 = 1 1 '-12 is an arbitrary partition of [S]2, then either there exists an S' C S, S' = 0, such that [S']2 C 71 or there exists an S" C S, S" = na such that [S"] 2 C I2 .



119

Classification of the denumerable order types

0

(01',„1)2

indicates the negation of this statement . The symbol just defined has the following obvious monotonicity properties

0-_, (0I , )tt) 2 O (017 m)2

implies

0'--,(01,

implies

0

(O í,

nt)2

7n')2

0 < 0' , for every 01 < 0 17 for every

111'

< m .

3 . Classification of the denumerable order types . Let S be an ordered set ordered by the relation R, and let a. b (R) be two arbitrary elements of S . (a, b)(R) denotes, as usual, the interval {x : x E S and a. < x < b(R)} . The ordered set S is said to be dense if (a, b) 0 for every pair a < b E S. The order type 0 is said to be a discrete type if S = 0 and S does not contain a dense subset . Let 4 denote the set of all denumerable order types and let ,D be the set of all discrete denumerable order types and put 4 s = 44D-Considering that every denumerable dense set is of type )j, 1=, q, yl + 1, or 1 + 77 + 1, the following statements are immediate consequences of the above definitions .

1)

3 .1 . If 0 0.

E

_J them 0

E

JD if awed only if

0 and 0

E

4 s if and only

Now we are going to define a class 0 of denumerable order types . DEFINITION 3 .2 . We define the classes 0„ for every o < co, by transfinite induction on o as follows . O o consists of 0 and 1 . Suppose that 0 e is defined for every o' < o for a o < w1 . Put G„ _ U 0 é . Let 0. consist of the o)-sums and of the co*-sums of the sequences 0 0 , . . ., 0,, . . . satisfying the condition 0,, E G, for every n < w . It is obvious that O a C . . . C Oo C . . . for o < 0' . Put 0 = J Oe . e o . But this means that S* is



122

P . Erdös and A . Hajnal

either of type 1 or dense, and-being denumerable-it is of type 1+-rt, or 7j+1, or 1-, .q-4-1) . -Tow we prove that

q

(or

3 .15 . There exists an ordinal number y, < w l such that S*o = S*p+l . Proof . By the definitions 3 .7 and 3 .11 corresponding to every

element x of S, S*(x) is a non decreasing sequence of subsets of S, and thus-S being denumerable-there exists a y (x) < wl such that S*(x) = S* for every > , , (x) . Using again the fact that S < d o we infer that there exists a y o < co, such that yo > y ( x) for every x S and consequently S*., (x) = AS,;o ~(x) for every x e S, whence S, * = S;*~ T . E

~

DEFINITION 3 .16 . By 3 .15 we can make the following definition . Let y(O) = y be the least ordinal number y < wl for which S* = S*+l . y(0) will be called the order of O . . It is obvious from the above considerations that So* is remark a: proper refinement of S* for every ( < a < y (0) and that S* (e ) = S.* for every y > y(O) . It follows that the sequence g!(O, y) is non-increasing (cp(O, ) < (O, for ; > y') but it is not strictly decreasing even if O is ,in ordinal number . For example, put O S = (ww) . Then (ww) = o-), w ) = 1 but V (c)w , for every integer By 3 .14 we have V( O, y (O)) =1, )7, 1--i) i ip 1 or 1+)j+1 . Considering that S.,**(x) 0 for every x x5', y < w l , IT (0, y(O)) T 1 implies ~~ O . It follows from 3 .1 that y. (o')w ,

n)

=

cow

E

3 .17 . -If O E 4 D then T (O, Y (O)) = 1 . Vow we need preliminaries concerning the class 0 . 0=

3 .18 . Suppose that Z = S O X E0 .

, ~VE 0, and O,

E

0 for every x

E

Z . Then

;f.EL

Proof : By induction on 0(9?) . The statement is obvious for p(T) = 0 . Suppose that it is true for every type T,' with P((p)' < P for a 0 < Q < wz . Then, by 3 .5, Z is either the co - sum or the o)* - sum of the sets Z n of type T,, of rank less than o . The types On = Z O„ then belong to 0 by the induction hypothesis XEZ,l

and O is either the

o) -sum

or the

co*

-sum of them, whence O

3 .19 . a, a*

E

0.

E 0 for every a < w, . Proof . By symmetry it is enough to prove this for a . We use induction on a . O .E 0 and if a > 0 then either a = # + I or a is of the second kind and consequently is cofinal with a) . Hence in both cases it is the co-sum of ordinals less than a which belong to 0 by the induction hypothesis . Now we are going to prove that

3 .20 .

JD C 0, 'T (0, y (O)) = 1 implies 0

E

0 for every 0 E 4 .



Classification of the denumerable order types

Proof . If O

123

d D then, by 3 .17, cp(O, y(O)) = 1 . We are going to

E

prove by induction on y (0) that T, (0, y (O)) = 1 implies O e 0 . If y (0) = 0 then, by 3 .13, S(R) = So (Ro) = 1, whence O = 1, O E 0 . Suppose that y (0) = y > 0, y < co, and that O' E 0 for every 0 provided y(O') < y and q? (0', y(O')) = 1 . We distinguish two cases : (i) y = # + 1, (ü) y is of the second kind . Ad (i) . S* = 1 (R*) . Hence S*(x) = S for every x E S. By 3 .13, S* is the splitting S" of $ induced by the splitting S;* (defined in 3 .11) and thus S = SY(x) = U SO(y) 8d(y) e N(S;(x»

It is obvious that the order of the sets S ;(y) ordered by R is < p < y, and thus S*(y) (R) belongs to 0 by the induction hypothesis . Considering that by 3 .12 O is the co - sum, the w* - sum, the w* + w - sum, or a finite sum of them, O belongs to 0 . Ad (ü) . S* = 1 (R*), whence by 3 .8 and 3 .13 S = S**(xo ) _

Sá(xo )

for an arbitrary fixed x,, e S . Considering that the order of every Ss(x) is < Í3 < a, we infer from the induction hypothesis that S,*(x,) (R) belongs to 0 for every # < a . Put A O _ {x : x

E

={x : x

E

BO

S and x < x, (R) and x

E

80(xo)- U S;*'(xo)J

S and x>x, and xES,3(xo )-USá(xo )f 91 «I) + 1, .;0 ) 2

(1)

provided

w - w* - 0 .

o (0) . It is that 0 is of the form easy to verify, for example by induction on g (0), By

w

•

w*

A 0, 0 is discrete and by Theorem 1 it has a rank

Y, pv , where a and (, (v < a) are ordinal numbers .

v