AbstractIn this paper we show the extent to which a finite tree of fixed height is a Ramsey object in the class of trees of the same height can be measured by its symmetry group

Fouché, W.L.

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ELSEVIER Discrete Mathematics 197/198 (1999) 325 330 
DISCRETE 
MATHEMATICS 
Symmetries and Ramsey properties of trees 
W.L .  Fouch6*  
Department of Mathematics' and Applied Mathematics. Unil,er~'itv ~?[ Pretoria, 
0002 Pretoria. South A/rica 
Received 9 July 1997: revised 11 June 1998: accepted 3 August 1998 
Abstract 
In this paper we show the extent to which a finite tree of  fixed height is a Ramsey object m 
the class of  trees of  the same height can be measured by its symmetry  group. @ I999 Elsevier 
Science B.V. All r ights reserved 
1. Introduction 
In the sequel, all the combinatorial structures referred to will be finite. For a natural 
number r, we write [r] for the set {1 . . . . .  r}. in this paper, a tree will be a poset T 
with a minimum element (the root) such that, for each x ~ T, the set {y C T: y<x} 
is a linear order. The cardinality of  this set is called the heiqht of x and denoted by 
ht(x). The height of a tree is the maximum height of  any element of  T. For h>~0, we 
denote the class of trees of height h by ,~,. If T~, T2 ~ ,~,, an embedding tl of  T~ into 
T2 is a poset embedding (i.e.,/x is injective and x <y  iff IL(x)<t~(y) for all x, y~ Tl) 
which is, in addition, height-preserving, i.e., htOl(x)) = ht(x) for all x ~ T~. For a tree 
T, the number of  successors of  an element x is called the arity of x and denoted by 
ar(x). We denote the set of copies Tt in ~ by [~,  T~]. If T is of height h, a subtree 
of T is an image under an embedding of some tree S of height h. 
For T ~ .~,, let A(T)  be the group of height-preserving automorphisms of T. Set 
k 
where n~ . . . . .  nx. are the arities of  the elements of T. For reasons that will soon become 
clear, we call r (T)  the Ramsev degree of T. 
We recall that if c~ is a class of  finite structures for which we have a notion of a 
copy (image under an embedding) of an object A in an object B of c6, then A is said 
* E-mail: wlfouche(~math.up.ac.za. 
0012-365X/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved 
PII: S0012-365X(98)00243-X  
326 l/EL. Fouchd/Discrete Mathematics 1971198 (1999) 325-330 
to be a Ramsey object in ¢g if for each B in ~ and r <co, there is some object C 
such that for each partition Z : [C,A] ~ [r], where [C,A] is the set of copies of  A in 
C, there is a copy B' of  B in C such that all the elements of  [B',A] are in one block 
of the partition. The aim of this paper is to show, as was done in [3,4] for posets and 
bipartite graphs, respectively, the extent to which an object T is a Ramsey object in 
3--h can be measured. Our main theorem explains how the number r(T) does it for a 
tree T. 
Theorem 1. For integers , h>~O and trees T, T1, both of height h, there is a tree T2 
of the same height such that, for every Z : [T2, T] -+ [s], there is a copy, T(, of 1"1 in T2 
such that Z assumes at most r(T) values on [T(, T]. Moreover, for a 9iven tree T of 
height h, and for r >~r(T), there is some T1 E ~-h, such that, for any tree T2 E ~-h with 
[1"2, T1] ~ (~, there is some partition Z : [T2, T] -~ [r] with the property that )~ assumes, 
on any copy of T~ in T2, at least r(T) values. 
It follows that T will be a Ramsey object in ~ iff r(T) = 1. A tree is complete when 
the arities of  any two elements of the same height are the same. We shall soon see 
that r(T)~- 1 iff T is complete. It follows that the complete trees are the only Ramsey 
objects in ~.  For some further results on the Ramsey properties of trees, the reader 
is referred to [1,2, 6,7]. An analogous result for distributive lattices appears in [8]. 
2. Proof of main theorem 
For each i/> 0, the set of  elements of  T of  height i is called the ith level of  T and 
is denoted by Li(T). For T E J-h, let T c be the complete tree such that, for every i<<,h 
the arities of all x ELi(T c) will be max{ar(x): x CLi(T)}. We call T c the completion 
of  T. An orientation of the tree T is a sequence O of the form (O1 . . . . .  Oh), where, for 
some embedding/~ of T into T ~, each Oi is the sequence of arities of/~(T), the image 
of T under /~, as they occur at level i of  T c, when we scan the latter from the left 
to the right. We call the pair (T,O) an oriented tree. We say in this case that the 
orientation O is induced by/~. (Note that a given orientation can be induced by more 
than one embedding of T into TC.) We can visualise (T, O) as the tree T drawn in the 
plane in such a manner that the arities of  the ith level of  T appear as in the sequence 
Oi when Li(T) is scanned from the left to the right. 
It is clear that each complete tree has exactly one orientation. Moreover, if a tree has 
a unique orientation, then it must be complete. For if T were not complete, consider the 
first level where there are elements x and y with ar (x )¢  at(y). Take any embedding 
# of T into T c. Let cr be an automorphism of T c that interchanges x and y. Clearly 
a# is another embedding of T into T c. It is clear that the orientations induced by # 
and a/~ are distinct. 
I f  T' is a subtree of  T, then T' inherits a unique orientation from any given orienta- 
tion O of T, namely, if O is induced by the embedding # : T ~ T c, say, the inherited 
I/EL Fouchb / Discrete Mathematics 197/198 (1999) 325~30 327 
orientation of T' is induced by the restriction of p to T'. Here it is important o note 
that the inherited orientation of T' depends on O only. I f  (T, O 1 ) and (S, 0 2 ) are ori- 
ented trees, an embedding #: (T, O 1 ) --, (S, O 2) is a tree embedding such that #(T) in- 
herits the orientation O 1 from the orientation O 2 of S. 
Lemma 1. For a given tree T, the Ramsey degree, r(T), is the number of distinct 
orientations of T. Hence r(T) = 1 iff T is complete. 
Proofi Let h be the height of T. The proof proceeds by induction on h, the case h = 0 
being trivial. Let T be a tree of height h. We can represent, for a given orientation 
O of T, the oriented tree (T,O) as a sequence (Tl,O (I)) . . . . .  (T/,O (f)) of (oriented) 
trees of height <h. Here TI,..., T~ are the components of the forest, such that, when 
viewed from the left to the right, (T, O) is exactly a root connected with the sequence 
(TI ,O (11) . . . . .  (T/,O (/)) of trees. Suppose that there are exactly r isomorphism types 
of trees among the T,- (now with their orientations ignored) with cti of them being of 
the ith isomorphism type, for i=  1 . . . . .  z. Let M be the multiset {1 x', . . . .  z~}. 
Each orientation of T is obtained by first finding orientations O I . . . . .  O / for T~ . . . . .  T,, 
respectively, and by then permuting the oriented trees (Ti, 01 ) . . . . .  (T/, 0 / ). Two such 
permutations yield the same orientation of T i f f  they induce the same multipermutation 
of M. Hence, by induction, writing s(T) for the number of orientations of T, we have 
s (T )=(#)  s (T ' ) ' ' ' s (T ' )~ l , . . . , c~ 
( ( ) r (T1) .  r(T/). 
9~ I . . . .  , O~r 
Moreover, it is easily seen that 
IA(T)t = ~1 ! " "  ~!  tA(T~)I.-. IA(T/)I. 
Consequently, s(T) = r(T). Finally, we have already shown that s(T) = 1 iff T is com- 
plete. 
If  A is a set and n~>0 we write ()4) for the set of n-subsets of a.  In particular, (/i) 
is the singleton {(~}. For integers nL .. . . .  n~, >>-0 and m,n,s>~ 1, we write 
n~[m,  (nl . . . . .  n~,)] 
when the following holds: 
I ,  For sets Y1 . . . . .  Y,, with IY/I ~>n, and an s-colouring qJ: [I i=, (nr',)-+ [s], 
there is, for each i E [v], some subset Zi of Yi, each set Zi having exactly m 
elements, such that ~k assumes a constant value on the set 1-Ii~. (zl). 
The existence of n, for given ni, m and s is a well-known result (the so-called product 
Ramsey theorem, see for example [5]). 
In the sequel, we shall think of complete trees as being represented in the plane. 
If  T1, T2 are complete trees of the same height, an embedding # : TI --+ T2 is said to be 
328 W.L. Fouchd/Discrete Mathematics 197/198 (1999) 325-330 
strong if, for each level Li of  TI, if x, y c Li and x is to the left of y then /z(x) will 
be to the left #(y)  on the ith level of  T2. We have introduced this idea in order to 
ensure that the induced orientations of subtrees of TI are preserved under/~. 
For trees T, Tl, T2 E ~--h where Ti and T2 are complete, for a natural number s and 
an orientation O of T we write 
T2 ~ [T1, (T, O)] 
when the following holds: 
For every s-colouring g of  the copies of  the oriented (T, O) in /2, there is a 
strong embedding /z:Tl ~ T2 such that Z assumes a constant value on all 
the copies of (T, O) in T2 which are contained in the image of Tl under/~. 
In order to prove Theorem 1 it suffices to prove 
Lemma 2. For h >>, 0, s >~ 1 and T, T1 E 5-:, with Ti complete, there exists, for every 
orientation 0 of T, a complete T2 E ~,  such that T2 ~ [T1, (T, O)]. 
We now deduce Theorem 1 from Lemma 2. Let O 1 . . . . .  O" with r=r (T )  be the 
distinct orientations of  T. Set TO= T~ and apply Lemma 2 to find complete trees 
T 1 .... , T" such that 
• s i T '+ :=~[T, (T,O')] 
for i ----- 0, ., r - 1. Then a simple downward induction shows that T2 = T" satisfies the 
conclusion of  Theorem 1. 
If (T, O) is an oriented tree and if S is complete, both of the same height, we 
write [S, (T, O)] for the set of copies of  T in S that inherit the orientation O from the 
(unique) orientation of S. 
We now prove Lemma 2 by an induction on h, the case h = 0 being trivial. Assume 
Lemma 2 holds for some h ~> 0. Let T, Tl E ~::,+l with T1 complete. Let T~ I) and T (l) 
be the first h levels of Tl and T, respectively. Write 0 (1) for the orientation of  T (~) 
which is the restriction of  O to the first h levels. If O = (O1 . . . . .  Oh+l ), say, we denote 
the sequence Oh by (nl . . . . .  n~,). 
It follows from the induction hypothesis that there is a complete T~ l ) E ~ such that 
~1)~ [T~I), (T(,) 0(,))]. (1) 
Enumerate the elements of  [T~I),(T(I),o(1))] as  T | . . . . .  T •. 
Set m0 =k,  the arity of  any element of  Lt,(TI ). Define ml . . . . .  m/, iteratively by 
mi+l ~ [(mi, (nl . . . . .  n,,)], 
i = 0 . . . . .  # -- 1. Let T2 be a complete tree of height h + 1 whose first h levels are 
exactly those of  T2 (1) and such that each element of the hth level of Te has arity me,. 
We shall show that T: has the properties as in the statement of Lemma 2. 
VKL. Fouch6/Discrete Mathematics 197/198 f 1999) 325 330 329 
Label the elements of  L/,(T2) as they appear from left to right by the numbers 
1 . . . . . .  [ .  In the sequel we shall identify Lh(T2) with [f] .  The sets of  successors of 
1 . . . . . .  /' in T: are denoted by X~ ). . . . .  X~ ~, respectively. 
For given Z: [T2,(T,O)]--+[s], we shall construct an induced colouring 0 of [QI,. 
(Ttt).o~l;)] in stages, one stage for each copy T i of(T I ILo/~t)  in T{ li. For i 0 . . . . .  I~. 
let P, be the following statement: 
There are subsets X~, .... X I o fX  ° . . . . .  X{l! each of size >~m .... , such that, tbr 
all extensions T" of T 1 .. . . .  T" to copies of  (T, O) in T~ such that L/, ~ I (T ' )  C 
U / I  x j ,  the colour of T" with respect o Z depends only on the copy, S, of 
T i~) of which it is an extension. 
The statement P0 is vacuously true. Next assume P, holds for some i with 0 ~<i</t. 
Let p (1 )<. . .  <p(v)  be the subsequence of [ f ]  (=Lh(T2)) which are the points at 
the hth level of T i : l .  Set Yi =Xj, ul and define 
() YI x . . .x  -+[s ]  I l l :  / I  I nt, 
as follows: For (~1 . . . . .  v,.) in (~,i) × " '×  (:'i)' let To be the copy of (T,O) which is 
the extension of  T t-I such that T/ is the set of  successors of p(j) ,  for j = I . . . . .  r. Set 
~(TI . . . . .  v,) /.(T0). Since 
m~, ,. ~ Imp, ci+,~,(nl . . . . .  n,.)], 
there are Z/C Yi with Iz/I = m,, u-I), J = 1 . . . . .  v, such that ~ is monochromatic on 
Set 
X!-' {Zk 
/ X/ 
(,0 
if j = p(k)  E L/,(T '-t ), 
otherwise. 
Then the sequence X{ +l , j = 1 . . . . . .  f witnesses the truth of Pi+~. 
Since p, holds, and m0 k, the number of successors of any xE Tj at level h. 
yt~ y l l  there are sets "'l . . . . . . .  /C  Lh+I(T2) with Ix,!'l ~>k for all j ,  such that all the extensions 
T" of an element S of [T~I),(TIII,Od))] such that L/,+,(T")C U~ ,x/ '  will have 
a colour that depends on S only. We have thus found an induced s-colouring 0 of  
[r~l I, (T~, Ot ~/)]. 
By the relation (1), there is a copy, R (1t, under a strong embedding of TI 1~ in T~ I~ 
such that 0 assumes a constant value on [R Ill, (T Ill, 0 I1~)]. Write q(1)< ..- < q(t) tbr 
the elements of  Lh(RII)). Let R be the extension of R/l~ to a complete tree of height 
h + 1 such that the successors of  q(i) are given by the set Xlli), i 1 . . . . .  t. Then R 
contains a copy of Tt and, moreover, Z is constant on [R,(T,O)]. This completes the 
proof of  Lemma 2 and hence of the first part of  Theorem 1. 
We now prove the second part of  Theorem 1: Choose T C .~, and set T~ T< Let 
T2 be any tree of height h with [~.Tt]:~(~. For T 'C[~,T] ,  set z(T ' )  O when 
330 W.L. Fouchb/Discrete Mathematics 197/198 (1999) 325-330 
O is the orientation that T t inherits from T~. Then Z assumes on any set of the form 
IT(, T], with T( a copy of T1 in T2, exactly r(T) values. 
Acknowledgements 
The author wishes to thank the (anonymous) referees for their constructive remarks 
on this paper. 
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[3] W.L. Fouchr, Symmetry and the Ramsey degree of posets, Discrete Math. 167/168 (1997) 309-315. 
[4] W.L. Fouchr, Symmetry and the Ramsey degrees of finite relational structures, J. Combin. Theory Ser. 
A, to appear. 
[5] R.L. Graham, B.L. Rothschild, J.L. Spencer, Ramsey Theory, Wiley, New York, 1990. 
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Symmetries and Ramsey properties of trees 

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Symmetries and Ramsey properties of trees

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