Discrete Mathematics
North-Holland
54 (1985) 339-341
339
COMMUNICATION
TWO REMARKS
ON RAMSEY'S THEOREM
Jaroslav N-ESETI~,IL and Vojt~ch RODL
Department of Mathematics, F3F1 ~VUT, 110 O0 Praha 1, Czechoslovakia
Communicated by T. Brylawski
Received 6 December 1984
We present a very simple proof of the fact (due to P. Erd~s and R. Rado) that Ram~ey's
theorem doesn't hold for partitions of infinite subsets. We also present a proof of an induced
Ramsey theorem for partitions of complete subgraphs (due to W. Deuber and authors) based
on the theorem of R. Graham and B. Rothschild on parameter sets.
latrodaetion
The purpose of this note is to give an alternative and simple proof to the two
particular Ramsey-type theorems: K-/-~(to)~' for any cardinal K (due to ErdSs and
Rado) and to the statement VG ::IHH---, (G)~, where both G and H are finite
graphs (due to W. Deuber and authors).
1. ~ ~ s
of ~ i t e
subsets
Recall that, given cardinals a, /3, % 6, The Erd6s-Rado partition arrow
--~ (a)~ has the following meaning: for every partition c :[3,]°--> 8 there exists
? t < 8 and a set X---7 [XI =,v, such that c ( Y ) = ) t for every YE[X] °. Similarly
7-7~ (a)sa denotes nonvalidity of this statement. As usual, [T] ~ = { A ; A _ 3' and
IAI=/3} and cardinal numbers are identified with initial ordinals. Ramsey's
theorem states to ~ (to)sa for every linite 8, /3 and Erd/Ss-Rado proved that
V/3 < to V8 V a =IT "y --~ (a)~ and that this fails to be true for/3 I> to. The last part of
this statement will be proved here.
Theorem I ([2]). (AC) V~/T-~(to~.
Proof. Assume the contrary: T - * (to)~'. Define the graph G = (V, E) as follows:
V = { Y ~ % 'the ordinal type of Y'=to}, { Y , Y ' } ~ E ill Y - Y ' = m i n Y. (Explicitly: Y and Y' form an edge if Y = {Yo, Yx, Y2,-..} and Y ' = {Yl, Y2,.- -} where
Yo< Yl<" " ").
It is easy to see that the graph G does not contain any cycle. Consequently, G
has the chromatic n u m b e r = 2 (here (AC) is used). On the other hand ~ / ~ (to)~'
implies X(G) > 2 (in fact x(G) ~ to). []
0012-365X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
J. Ne~et~il, V. R6dl
340
2. Psrtitions of complete subgr~hs
Denote by K~ the complete graph with p vertices. For graphs H, G and positive
integers k, p write
H--* (G)~:,
if the following statement holds: For every partition ( ~ ) = A1 U . - . U Ak there
exists an induced subgraph G' of H, G ' - - G, such that (~) c_ Ai for an i ~ k. (Here
(~) is the set of all subgraphs of H which are isomorphic t o / ~ . ) We present here
a simple proof of the following:
T h e o r e m 2. For every positive integers k and p and for every graph G there exists a
graph H such that
This theorem was proved in [1] and [4] by means of direct construction.
Another proof is given in [5] where a much more general theorem with a
complicated proof is given.
First we introduce some necessary notions: Given a finite set X and positive
integer p define (~) as follows:
Ai n A i = 0 for every i ¢ ].
If X = { 0 , . . . , n - 1 } then we write shortly (~. We make use of the following
theorem which is a special case of the theorem proved in [3]:
Theorem 3. For every positive integers k, p and n there exists a set X with the
following property: For every partition (fx) = ~ 1 ~,-j" " • [..j " ~ k
{ 1 , . . . , k} a n d sets Y1, . . . , Y,,, { Y1, . . . , Y,,} e (~c) such that
{/U
EAt
i~A2
there exists an i
i~Ap
for every ( A a , . . . , A p } e ( ~ ) .
Proof of Theorem 2. It is a well-known fact that for every graph G = (V, E)
there exists a set X and a 1-1 mapping f : V - , P(X) such that
{v, v'}eE
Denote by •(X) the graph (P(X), {{M, Mr}; M ClM ' = 0}). Consequently, it
suffices to prove that for every set X and for every positive integers k, p there
exists a set Y such that
a(v)--,
Two remarks on Ramsey's theorem
341
However, as the sets (n~)) and (~,) are 1-1 correspondence this follows immediately from Theorem 3.
Retereaees
[ 1] W. Deuber, Pattitionstheoreme fiir Graphen, Comm. ]Vlath. Helvetici 50 (1975) 311-350.
[2] P. Erd6s and R. Rado, a partition calculus in set theory, Bull. Amer. Math. Soc. 62, 427-489.
[3] R.L. Graham and B.L. Rothschild, Ramsey's theorem for n-parameter sets, Trans. AMS 159
(1971) 257-292.
[4] J. Ne~et~ and V. R/~tl, Partitions of subgraphs in: M. Fiedler, ed., Recent Advances in Graph
Theory (Prague, Academia, 1975) 413--423.
[5 ] J. Ne~efffl and V. R6dl, Partitions of finite relational and set systems, J. Comb. Theory Set. A 22
(3) (1977) 289-312.