AbstractWe will prove that for each n ≥ 3 the homogeneous Kn-free graph Hn is indivisible. That means that for every partition of Hn into two classes R and B there is an isomorphic copy of Hn in R or in B. This extends a result of Komjáth and Rödl [Graphs Combin., 2 (1986), 55–60] who have shown that H3 is indivisible

El-Zahar, M

Sauer, N

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JOURNAL OF COMBINATORIAL THEORY, Series B 47, 162-170 (1989) 
The Indivisibility of the Homogeneous &-Free Graphs 
M. EL-ZAHAR AND N. SAUER* 
Department of Mathematics and Statistics, University of Calgary, 
Alta, Canada T2N IN4 
Communicated by the Editors 
Received January 5, 1987 
We will prove that for each n 2 3 the homogeneous &-free graph H, is 
indivisible. That means that for every partition of H, into two classes R and B there 
is an isomorphic copy of H, in R or in B. This extends a result of Komjath and 
Rod1 [Graphs Combin. 2 (1986), 55-601 who have shown that H, is indivisible. 
0 1989 Academic Press, Inc. 
INTRODUCTION 
Graphs are undirected and without loops. The graph A is a subgraph of 
the graph G if V(A) c V(G) and E(A) = (A x. A) A E(G). A countable graph 
H is homogeneous if for every two finite subgraphs A, B of H and every 
isomorphism f: A -+ B there is an automorphism of H which extends J 
This definition follows Frays& [2, p. 3131. Lachlan and Woodrow [3] 
named those graphs “ultra-homogeneous.” In [3] they characterised the 
homogeneous graphs. 
We will rely on this classification of the homogeneous graphs and on the 
more general information in [2]. A graph H is indivisible if for every parti- 
tion of the vertex set of H into two classes R and B there is an embedding 
f: H -+ H such that f(H) c R or f(H) c B. (See [2, 0. 1741.) 
It is clear that the complement of a homogeneous graph is again 
homogeneous and that the complement of an indivisible graph is 
indivisible. Lachlan and Woodrow [3] proved that up to taking com- 
plements the only homogeneous graphs are disjoint copies of isomorphic 
complete graphs, the universal graph @ and the &-free homogeneous 
graphs H, (n > 3). Of the graphs in the first group only K, and the graph 
which consists of cc) copies of K, are indivisible. That the random graph @ 
is indivisible can easily be seen, but also follows from [2; 10, 4.4, p. 2881. 
In [ 1 ] it is shown that H3 is indivisible. In this paper we will now prove: 
THEOREM. Every homogenous &-free graph H, (n 2 3) is indivisible. 
* This work was supported by N.S.E.R.C. Grant 69-1325. 
162 
0095-8956/89 $3.00 
Copyright 0 1989 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
HOMOGENEOUSK,-FREEGRAPHS 163 
1. SOME PRELIMINARY OBSERVATIONS 
We assume familiarity with the description of H, given in [4]. 
Familiarity with the proof that H, is indivisible as presented in [ 1 ] is not 
strictly necessary but probably helpful. 
LEMMA 1. Let G be a finite K-free graph and x E G. Then to each 
embedding f: G - x + H, there are infinitely many extensions off to an 
embedding of G into H,. 
Proof There is an embedding g: G -+ H, [4, Theorem 2.31. f 0 g-l: 
g(G)-dxhfw- ) x is an isomorphism and can therefore be extended 
to an automorphism CT of H,. cr 0 g is then an extension of f: If G is a 
&-free graph with V(G) = (x,, x2, . . . . x,=x>, then the graph F with 
V(F) = VW u (Y>, no edge from x to y, y is connected to Xi for 1 < i < n 
just in case x is connected to xi, is also K,-free. This observation implies 
that we can extend f in infinitely many ways. 1 
LEMMA 2. The K-free graph A is isomorphic to H, if and only if: 
for every finite &-free graph G and x E G each embedding 
f: G - x -+ A can be extended to an embedding of G into A. 
Proof Follows from Theorem 2.3 of [4]. 1 
COROLLARY 1. If XC V(H,) is finite then H - X is isomorphic to H,. 
([4, Corollary 2.51.) 
COROLLARY 2. Let X be a finite subset of V(H,). Let Y = { y E H, - X; 
(Y9 4 #WC) f or every XE X]. Then the subgraph induced by Y is 
isomorphic to H,. 
LEMMA 3. Let G be a finite I&-free graph and XC V(H,), such that G 
cannot be embedded into X. Then H, - X contains an isomorphic copy of H,. 
Proof This is Theorem 4.1 of [4]. 1 
2. PROOF OF THE THEOREM 
We will prove the following: 
THEOREM. The K,,-free homogeneous graph H, is indivisible (for all 
natural numbers n 2 3). 
164 EL-ZAHAR AND SAUER 
Proof: We shall assume that H, is defined on the set of natural 
numbers. We will use the following notation 
X< Y++ maxX<min Y tx ycv 
a<Y+-+ (a}<Y (aEN, YcN) 
[+-a]={xEN:x<a} 
[+-a)=(xEN:x<a}, 
etc. 
DEFINITION. For a E H,, r(a)= (x~H,:x<a and xaEE(H,)). The 
vertex a will be called a strong vertex if r(a) is nonempty and contains the 
complete graph K, _ *. Otherwise a is a weak vertex. 
LEMMA 4. The set of strong vertices of H, contains an isomorphic copy 
of& 
Proof The set of all weak vertices cannot contain K,- i. Apply 
Lemma 3. 1 
DEFINITION. Let a E H, be a strong vertex. We define 
yl(a) = min r(a). 
r,(a) = min{x: r(a) n [ + X] contains Ki) for 2<i<n-2. 
rnwl(a) = min{x: r(a) c [ +- x)>. 
r,(a) = (X E r(a): y,(a) <X < yi+ ,(a)} for 1 <i<n-2. 
Note that yn _ I (a) 4 r(a). We could even have yn _ l(a) = a. 
In what follows we shall assume that the vertices of H, are colored, in 
an arbitrary but fixed way, with two colors red and blue. We also introduce 
another (uncolored) copy H,* isomorphic to H,. The vertices of H,* will be 
denoted by a,, a2, . . . and we assume that they are ordered such that 
a, <a2< .... We do not assume that the graph and order isomorphisms 
between H, and H,* coincide. Since the vertices of H,* are ordered, we can 
define strong and weak vertices, Yi(a,), ri(a,) - - a, etc. in H,* . 
Statement. H, contains a monochromatic copy of itself. 
The statement will be proved by constructing a monotone map 
0: Hz -+ H, with the following properties: 
HOMOGENE~USK,-FREE GRAPHS 165 
1. The set { ~(a,): ak is a strong vertex of H,* ) is monochromatic. 
2. For every pair of natural numbers, i, j, i < j, we have 
4ai) 4aj) E E(H,) ++ aj is a strong vertex of H,?f and 
aiajfE(H,1:). 
For every vertex x E H, we define two formulas: the blue formula q(x) and 
the red formula t,G(x). The blue formula q(x) is the following formula: 
(3F, 3x, : x = minF, A F1<al A K2 GL F,) 
(VE,V'~,:~,<E,<&AK, dF,uE,) 
(3F2 3~~:~~ -+,<a, A K, CL F1 uE, uF,) 
(h?2~/?2:a2<E2<~2~& d~F,uE,uF2~E2) 
(3Fi3Ui:Pim1<Fi<al A Ki+, d F,UE,U .a. UEi-1UFi) 
(VEiV’Pi: ai< Ei<Bi A Ki.1 d F, UE, U **a UFiUEi) 
(3F,-, 3x,-,: /!$+3 <Fn-2 < ctnp2 A Kn-l (f F1 u E, u ... u En-, u Fnp2) 
(VE,-2V~n-2:an-2<En-2</.ln-2A K,-,~F,~E,~+JF,-,~E,,-~). 
(The set bwl: v>Pn-2 and T(y)n[tfl,-2]=F1uE1u -u 
F n-‘UK-2) contains infinitely many blue vertices.) V(X) is interpreted 
according to the following rules: 
1. : means “such that.” 
2. I;,, E, , . . . . F, _ 2, E,- 2 are restricted to be finite nonempty subsets 
of H,, and a,, P1, . . . . Lo, Pne2 are restricted to be elements of H,. 
The red formula $(x) is obtained from q(x) by interchanging the quan- 
tifiers V, 3 and replacing the word “blue” by “red.” One can convince 
oneself that for every XE H, at least one of q(x) and $(x) is true. 
DEFINITION. A vertex x E H, is called a blue generator if q(x) is true. 
Otherwise x is called a red generator. 
We shall divide the proof into two cases: The blue case where there are 
infinitely many blue generators in H, and the Red Case where there are 
only finitely many blue generators. 
166 EL-ZAHAR AND SAUER 
The Blue Case 
Here the formula cp will be used to generate certain subsets as follows. 
Suppose we have just chosen a vertex x E H, which is a blue generator to 
be included in our construction. Then, since q(x) is true, there exist 
F, c H, and a, E H, such that x = min F, and F, < a. We say that F1 and 
CI~ have been introduced and denote them respectively by F1 (x) and a,(x). 
We also say that the quantifier block 
cpl(X) = Wl w 1:G+a -01 A K2 Lf w+JEl) 
has been activated. If at a later stage some set E, c H, and an element 
p1 E H, were recognized to satisfy 
ca-4 < El < Pl and K2 (f w+JEl 
then E, , pi will be called an instance pair for cp i(x). Corresponding to E, , 
pi there exist F2 c H, and ct2 E H, such that 
Pl <F* < a2 and K3 & F,(x)uE, u F2. 
We shall denote these F2 and a2 by F2(x; E,: pi) and a,(~; E,; PI). In turn, 
the quantifier block 
(p&; E, ; pl) = (W v’p 2:~2(-w,;p,)<E,<p, 
* K, d F,(x) u E, u F2k E, ; PA u ~52) 
will be activated. 
In general, if the quantifier block 
cpik El, ***, Ei- 1; Pl, --*, pi- 1) 
-(vEy’Pi:ai(x;E, )..., E,-,;p, )...) Pi-l) 
<Ei<& A Kj+l d F,(x) uE, u 
. . . u&(x;& ,..., Eipl;& ,..., &ljuEi) 
has been activated, then a pair E,, pi satisfying this quantifier block will be 
called an instance pair for it. This paper will induce the existence of 
which in turn will activate the quantifier block 
HOMOGENEOUS K,-FREEGRAPHS 167 
Note that all the previous quantifier blocks 
are assumed to remain activated. In other words we can use them to 
generate sets as many times as we want. The sets E,, E,, . . . will be called 
initiators and the sets F,(x), F2(x; El, PI), . . . will be called the induced sets. 
The general picture of our construction will be as follows. There will be 
an infinite sequence of blue generators B, , BZ, . . . and another sequence 
consisting of ~(a,), ~(a~), . . . . We shall use sets of the form 
Ei= (~(~i,), ...) ~(aik)) as initiators to generate sets of the form 
f”(B,; El, *a*; PI 3 *a*) as induced sets. The relative position of these elements 
and sets will be as follows. Above a generator B, will be the set F1(Bk). 
This will be followed by ~(a,). Above ~(a,) there will be some induced sets 
of the form Fi(B,; Ei,, . . . . rRi,, . ..). i 2 2 and I < k. Next will lie the generator 
B k + 1. The open interval (~(a, _ 1), Bk) will be denoted by Zk. Except for the 
sets F,( Bj), each induced set will lie in an interval Ik for some k. All the 
induced sets will be so constructed to be disjoint and to contain no element 
BJ or ~(a,). In order to be able to do this we shall introduce the time 
bound T. At any stage of the construction the current value of T will be 
equal to some element in H, larger than all the previously constructed sets 
and elements. Having introduced the next element or set into our construc- 
tion, we set the current value of T to some element in H, which is above 
the newly introduced element or set and so on. 
Let us now describe our construction in full detail. 
Choose a blue generator B, and introduce B, and F,(B,). Set the current 
time bound T= a, (B, ). Choose a vertex y such that 
Y>T and T(Yb c 4- Tl = 0. 
Put ~(a,) = y and set T= ~(a,)+ 1. Now the pair ((~(a,)}, T) is an 
instance pair for the activated subformula cpl(B,) so that I;;(B, ; 
(4a,)); T) and dB,; (44); T) exist. We introduce F2( B, ; { ~(a,)}; T) 
and set T=a,(B,; {o(q)}; T). 
Next we choose a blue generator B, such that B, > T. We introduce B, 
and F1(B2) and set T=a,(B,). 
Let us now describe the construction after ~(a,- 1) has been introduced 
and the current time bound has been set to T = ~(a, _ 1) + 1. Consider all 
pairs ( Y, qi(Bj)) satisfying: 
6) Yc (+4, . . . . 44-d>, YZ 0. 
(ii) pi is an activated quantifier block belonging to Bj. 
(iii) for any T’E H,, T’ > T, the pair Y, T’ is an instance pair for 
CPA Bj)* 
168 EL-ZAHAR AND SAUER 
Order all such pairs in an arbitrary way. Pick the first such pair Y, 
Cpi( Bj). Since Y, T is an instance pair for cp,(Bj), we introduce 
Fi+ ,(Bj; . ..) Y; . ..) T) and set T=(x~+~(B~; . . . . Y; . . . . T). We repeat the same 
process one at a time for every such pair Y, cpi(Bj). Then we choose a blue 
generator Bk > T. We introduce FI(Bk) and set T= a,(B,). 
The next step is to show how ~(a,) is constructed. If ak is a weak vertex 
of H,*, then choose a vertex y E H, satisfying 
Y>T and T(Yb c 4- Tl = 0. 
We put a(ak) = y and set T = a(ak) + 1. Assume now that ak is a strong 
vertex of Hz. Define m, , . . . . m, _ 1 by 
Yitak) = arn, (in I-W, i = 1, . . . . n - 1. 
We also denote 
Di = dri(ak)), i=l , . . . . n -. 2. 
We claim that the subformula (P,, _ Z&1 ; D, , . . . . D,- 2; T,, . . . . T, _ 2) has 
been activated where Ti is a value of the time bound satisfying Ti E Im,+, 
(i = 1, . . . . n - 2). Assume for the moment that this is true. This means that 
FI(Bm,), F~(B,,; D1; T,), *.*F,(B,,; D,, . . . . Dip 1; T,, . . . . Tim 1), . . . have been 
introduced. From the formula cp(B,,), we deduce that there are infinitely 
many blue vertices y such that 
Y>L, and 
T(Y)~C~T,-,I=F,(B,,)UD,~F,(B,;..) 
u . . . uF,-,( -+JD,-~, 
choose such a vertex y satisfying y > T and put a(ak) = y. We also set 
T= c(ak) + 1. 
Assume now that all a(a,), 1~ k, have been constructed according to the 
previous description. We shall use induction on i to prove that all 
Vi(Brn, 3 D, 7 “-7 Di, 7’1, ***, Ti) have been activated where i < n - 2 and 
TjE1m,+l (j < i). First, if ai E rl(ak) then there is no edge aiam, E H,* . 
Therefore, according to the prescribed construction, F,(B,,) u D, is an 
independent set. Thus q2(Bm, ; D, ; T,) was activated and F,(B,, ; D, ; T,) 
was introduced when T = T, where T1 E Im2. 
Assume that for all j 6 i, the sets Fj(Bm, ; D,, . . . . Dj- 1 ; T1, . . . . Tj- I) have 
been introduced and the formulas 4oj( Bm, ; D1, . . . . Dj- I ; T, , . . . . Tj- 1) have 
been activated. For simplicity we shall write F, , F2, . . . instead of 
F,(B,,), F,(B,,; D,; T,), . . . . We also write cpl, (p2, . . . instead of ql(Bm,), 
HOMOGENEOU~K,-FREE GRAPHS 169 
‘PA&?, ; 4 ; w, *-* We show that Di, Tj is an instance pair for (pi where 
TiEI,,+~s We have to prove that the set 
X=F,uD,u -*. UFiUDi 
contains no Ki+ 1. Assume, on the contrary, that X contains vertices 
x1 < x2 < - . . < Xi+ 1 which form a Ki + 1. By the definition of I;i we have 
SO that xi+1 E Di. We also observe that (x1, . . . . Xi+ 1 > n UJ= 1 Fi # 0 since 
Ki + 1 & Uf= 1 r’( a,). Let h be maximum such that xh E Uj =, Fi, say 
xh E FP, p d i. Write x1 = ~(a~,) for h -C I,< i+ 1. We observe that h 6 p since 
K p+l & F,uD,u e-- u FP. Now each xz (h < I) has an edge to FP. By our 
construction, this implies that (yp(ak) > u rl (a,) u . . . u rP _ I(a,) for each 
h < 1. This is a contradiction, however, since D, u . - - u Tl(ak) u . - - u 
rP- Ja,) contains a complete graph K,, which together with aYh+,, . . . . ay,+* 
will form a Ki+ 1 in uf= 1 Tj(ak). This shows that Fi+ 1 has been introduced 
and cpi+ 1 has been activated or, in the case i= n - 2, we can introduce 
~(a,). This completes the proof in the blue case. 
The Red Case 
In this case, we can assume that each vertex in H,, is a red generator 
since deleting finitely many vertices of H, we still have a copy isomorphic 
to H, (in the graph-theoretic sense). The construction will go along similar 
lines as in the blue case except for some minor changes which we describe 
below. First of all we do not introduce special vertices to act as red 
generators since each vertex by itself is a red generator. We shall use the 
red formula to generate sets in the same way as we did in the blue case. 
The initiatiors, here the Fi)s, will be again sets of the form { ~(a,,), . . . . 
c(a,)}. For each vertex of the form ~(a,) the subformula 
tw(~k)).= (W v a, : ~(a,) = min F, A F, <a, A K, Ct F,) 
will be automatically activated. If the subformula 
170 EL-ZAHAR AND SAUER 
has been activated and Fi, ai is an instance pair of it then we can introduce 
the new set Ei(a(a,); F, , . . . . Fi; aI, . . . . ai) and the next quantifier block will 
be activated. 
The time bound T will act in the same way as in the blue case. Also the 
proof for the existence of ~(a,) for each k will be similar as in the blue case. 
REFERENCES 
1. P. KOMJATH AND V. R~DL, Coloring of universal graphs, Graphs Combin. 2 (1986), 55-60. 
2. R. FRA~SS~, Theory of relations, Stud. Logic Foundations Math. 118 (1986). 
3. A. H. LACHLAN AND R. E. WOODROW, Countable ultrahomogeneous undirected graphs, 
Trans. Amer. Math. Sot. 262 (1980), 5 l-94. 
4. C. W. HENSON, A family of countable homogeneous graphs, Pacific J. Math. 38, No. 1 
(1971). 


The indivisibility of the homogeneous Kn-free graphs 

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The indivisibility of the homogeneous Kn-free graphs

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