AbstractIn this paper, some properties of Ramsey numbers are studied, and the following results are presented. 1.(1) For any positive integers k1, k2, …, km l1, l2, …, lm (m > 1), we have r ∏i=1m ki + 1, ∏i=1m li + 1 ≥ ∏i=1m [ r (ki + 1,li + 1) − 1] + 1.2.(2) For any positive integers k1, k2, …, km, l1, l2, …, ln , we have r ∑i=1m ki + 1, ∑j=1n lj + 1 ≥ ∑i=1m∑j=1n r (ki + 1,lj + 1) − mn + 1. Based on the known results of Ramsey numbers, some results of upper bounds and lower bounds of Ramsey numbers can be directly derived by those properties

Li, Jinwen

Liu, Linzhong

Song, Enmin

Zhang, Zhongfu

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Applied Mathematics Letters 16 (2003) 1187-1193 
Mathematics 
Letters 
www.elsevier.com/locate/aml 
Some Properties of Ramsey Numbers 
ZHONGFU ZHANG 
Institute of Applied Mathematics, Lanzhou Railway Institute 
Lanzhou Gansu, 730070, P.R. China 
LINZHONG LIU* 
Department of Traffic and Transportation Engineering 
Lanzhou Railway Institute 
Lanzhou Gansu, 730070, P.R. China 
LLZH@public.lz.gs.cn 
JINWEN LI 
Institute of Applied Mathematics, Lanzhou Railway Institute 
Lanzhou Gansu, 730070, P.R. China 
ENMIN SONG 
Department of Computer, Huazhong Polytechnical University 
Wuhan 430074, P.R. China 
(Received May 2001; revised and accepted February 2003) 
Abstract-h this paper, some properties of Ramsey numbers are studied, and the following 
results are presented. 
(1) For any positive integers /cl, kz, . , km, 11,12,. ,1, (m > l), we have 
r (ii ki + 12 ii 4 + 1) 2 fi [r (ki + 1, zi + 1) - l] + 1, i=l i=l i=l 
(2) For any positive integers kl, kz,. . . , km, 11,12,. , I,, we have 
r (~k~+l~~~j+l) >~i:r(ki+l,~j+l)-~~+l. 
i=l j=l i=lj=l 
Based on the known results of Ramsey numbers, some results of upper bounds and lower bounds 
of Ramsey numbers can be directly derived by those properties. @ 2003 Elsevier Ltd. All rights 
reserved. 
KeyWords-Ramsey number, Clique, Independent set, Ingenuity graph. 
This research is supported by the NNSF of China (No. 19871036) and Qinglan Talent funds of Lanzhou Railway 
Institute. 
The authors are very grateful to the referees and Ynsheng Li for their helpful suggestions. 
*Author to whom all correspondence should be addressed. 
08939659/03/$ - see front matter @ 2003 Elsevier Ltd. All rights reserved. 
doi: 10.1016/S0893-9659(03)00176-9 
Typeset by d&-TBX 
1188 2. ZHANG et al. 
1. INTRODUCTION 
DEFINITION 1.1. For any given positive integers k and I, the Ramsey number r(k,l) is the 
smallest integer that makes each graph with at least r(k, I) vertices contain a clique with k 
vertices or an independent set with 1 vertices. 
It has been generally accepted that it is difficult to determine the Ramsey numbers. So it is 
meaningful to study the properties of the Ramsey numbers and their upper bounds and lower 
bounds. 
DEFINITION 1.2. If neither a clique with k vertices nor an independent set with 1 vertices exists 
in graph G, then G is called a.(k, I)-ingenuity graph. 
According to Definitions 1.1 and 1.2, the following lemmas are obviously true. 
LEMMA 1.1. Ifgraph G is a (k,Z)-ingenuitygraph oforder m, thenr(k,I) > m. 
LEMMA 1.2. If m < r(k, 1), then there exists a (k, l)-ingenuity graph with m vertices. 
The other terminologies and known results can be found in [l-4]. 
2. MAIN RESULTS 
First, we present a proposition as follows. 
PROPOSITION 1. For any positive integers kl, kz, 11,12, we have 
PROOF. Let Gi(V,E), i = 1,2, be two ingenuity graphs with r(kl + 1,11 + 1) - 1 and r(k2 + 1, 
12 + 1) - 1 vertices, respectively, then 
w(G) I ki, w (ci) I k, i = 1,2. 
Let V(G1) = {v~,vz, . . . ,z)~}, V(G2) = (~1~~2,. . ,un}, and quote Gi, j = 1,2,. . . ,m, denote m 
isomorphic graphs of G2 with vertices sets V(G$ = {ui, ui, . . . , uh}, j = 1,2, . . . , m, respectively, 
where uf maps to ur under the isomorphic mappings. We now define a graph G(V, E) as follows: 
V(G)= ijV(G;), 
j=l 
where (uf,uj”) E E’ iff (‘ui,uj) E E(G1). Ob viously, we have w(G) 5 klka, w(c) < 1112. Hence, 
r (hkz + 1,412 + 1) - 1 2 IV (G)I = (r (ICI + 1,11 + 1) - 1) (r (k2 i- 1,12 + 1) - 1)) 
which is the conclusion of this proposition. 
THEOREM 2.1. For any positive integers kl, kz, . . , k,, 11, 12,. . . , 1, (m > I), we have 
r (fix,+l.iil,+l) zfi[r(ki+1,li+1)-1]+1 
i=l i=l i=l 
PROOF. Use induction on m. If m = 1, then r(kI + 1, l1 + 1) 2 r(kl + 1,11 + 1) - 1 + 1, and the 
conclusion is true. If m = 2, according to Proposition 1, the conclusion is also true. 
Suppose the conclusion is true for any integer less than m. Particularly, for m. - 1 (m > 2), 
the conclusion is true, that is, 
/m-l m-1 \ m-1 
?- [I-f ki + 1, n li + 1) L n [r (ki + l,& + 1) - 11 + 1. 
i=l i=l i=l 
Some Properties of Ramsey Numbers 1189 
We now prove the conclusion is true for m: 
m-1 
2 n IT (Ici + 17 li + 1) - l][r (km + 1,1, + 1) - l] + 1 
i=l 
=II[~(ki+l,li+l)-l)+l. 
i=l 
So the conclusion is true. 
If ki = 11 = k, kz = 12 = 1, then we can obtain following corollaries immediately. 
COROLLARY 2.2. [See [l].) F or any positive integers k and 1, we have 
T (kl + 1, kl + 1) - 1 2 (r (k + 1, k + 1) - 1) (r (1 + 1,l + 1) - 1) . 
COROLLARY 2.3. If r(k + 1, k+ 1) = m + 1, for all n > 0, we have 
T (kn + 1, k” + 1) > mn + 1. 
PROOF. When n = 0, the conclusion is true obviously. 
When n > 0, by Theorem 2.1, we have 
r(kn+l,kn+l)>fi(r(k+l,k+l)-l)+l=m”+l. 
i=l 
By Corollary 2.3, we can obtain the following corollary immediately. 
COROLLARY 2.4. (See 111.) For all n > 0, we have 
r(2n+1,2n+1) >5”+1. 
PROPOSITION 2. For any positive integers ICI, k2, 11,12, we have 
4f 1. 
PROOF. Let Gi, i = 1,2,. . . ,4, be ingenuity graphs with 
IV(G) I = T (h + ~11 + 1) - I, tV(G2) I = v- (h + I, 12 + 1) - I, 
IV(G3)I=~(k2+1,ll+l)-1, IV (G4) I = T- (kz + ~12 + 1) - 1. 
. 
We have 
w (G1) I kl, w (&) 5 11; w (Gz) I kl, w (‘52) I: 12; 
w (G3) I kz, w ((73) 5 11; w (G4) 5 kzr w (e4) 512. 
1190 Z. ZHANG et al. 
Let a E V(Gl), b E V(G*), c E V(Gz), and d E V(G3). We now construct a graph as follows: 
V(G) = bV(Gi); E(G)=E’u ()E(Gi) , 
i=l ( ) i=l 
where E’ = {(u,v) ) u E V(G1) and v E V(Gs), or u E V(G2) and 21 E V(G4)) U {c&cd}. If a 
graph Go is obtained by joining any vertices between two vertices disjoint graphs G’ and G”, it 
is clear that w(Go) 5 w(G’) + w(G”) and w((&) 5 max{w((?), w(@)}, hence, we have w(G) 5 
kl+kz, w(G) 5 h+h, hence, 7-(h+h+1,~1+h+l)-l 2 jV(Gl)I+IV(G2)I+IV(G,)I+IV(G4)/, 
which is the conclusion of this proposition. 
When m = 2 and n = 1 (or m = 1 and n = 2), let G1 and G2 be two ingenuity graphs with 
IV(Gl)/ =r(kl+1,11+1)-1 andIV(G2)l =r(kz+l,Il+l)-1 (or IV(Gl)l =r(kl+l,Il+l)-1 
and /V(G2)/ = r(kl + 1,/z + 1) - l), respectively. Graph G is obtained by joining all of the 
vertices (or adding just one edge) between V(G1) and V(G2). Clearly, the conclusion is true. 
When m = 1 or n = 1, the conclusion is trivial. 
THEOREM 2.5. For any positive integers kl, k2,. . . , k,, 11, la,. . , l,, we have 
r (gki+l,klj+lJ ~~~r(ki+l,l~+l)-mn+l. 
i=l j=l i=l jzl 
PROOF. Combining with Proposition 2, we use induction on n. 
If n = 1, the following conclusion can be proved: 
>$fJr(ki,ll+I)-m+l 
i=l 
We now use induction on m. 
(1) If m = 1, then r(k1 + 1,11 + 1) = r(kl + 1, l1 + 1) - 1 + 1, so the conclusion is obviously 
true. 
(2) Assume the conclusion is true for all integers less than m. Without loss of generality, 
assume that the conclusion holds for m - 1 (m 2 2), that is, 
m-1 m-1 
T xki+l,ll+l 2 ~r(ki+1,11+1)-(m--1)+1. 
i=l i=l 
We now prove the conclusion is true for m: 
m-l 
xki+k,,,+l,ll+l 
i=l 
m-1 
IT ~ki+l,ll+l +r(k,+l,ll+l)-2+1 
i=l ) 
m-1 
2 c r (h + 1,11 + 1) + r (km + I,11 + 1) - (m - 1) -t 1 - 2 + 1 
i=l 
=gT(k;+l, ll+l)-mfl. 
i=l 
By the induction principle, the conclusion is true for any m 2 1 and n = 1, that is, 
> 
Z~~(ki+l,ll+I)-m+l. 
i=l 
Some Properties of Ramsey Numbers 1191 
(3) Assume the conclusion is true for all integers less than n, particularly for n - 1 (n 2 2), 
that is, 
T (,l;f,l,Fl,,l) ~~~r(ki+l,lj+l)-77(n-1)+1. 
i=l j=l i=l j=l 
We now prove the conclusion is true for n: 
m n-l 
2 c c r (ki + 1, lj + 1) - m (n - 1) + 1 
ix1 j=l 
+~r(ki+l,l,,+l)-m+I-1 
i=l 
=~~r(k,+1,!,+l)-mn+l. 
i=l j=l 
Combining the above process, the conclusion holds for any integers m > 1, n > 1, that is, 
r 
COROLLARY 2.6. 
PROOF. Let ki = 
have 
(gki+I,kl,+l) >~~r(ki+l,lj+l)--mn+l. 
i=l j=l i=l j=l 
For any positive integers k and 1, we have 
r (k,l) 1 (k - 1) (1 - 1) + 1. 
l,i=l,2 ,..., k-l,lj=l,j=1,2 ,..., 1-l,thenfromTheorem2.5,we 
r (k, 1) = T 
>xxr(2,2)-(k-1)(1-1)+1 
i=l j=l 
= 2 (k - 1) (1 - 1) - (k - 1) (1 - 1) + 1 = (k - 1) (1 - 1) + 1. 
From Corollary 2.6, it is easy to obtain following corollary. 
COROLLARY 2.7. For any positive integer k, we have r(k, k) 2 (k - 1)2 + 1 
COROLLARY 2.8. For any positive integers k and 1, we have 
r (2k + 1,21 + 1) > 5kl + 1. 
PROOF. By the above theorems and corollaries, we have 
r(2k+l,21+1)>(r(2+1,2+1)-l)(r(k+l,l+l)-l)+l 
> (r (3,3) - 1) kl + 1 = 5kl+ 1. 
1192 2. SANG et al. 
THEOREM 2.9. Every vertex in a (k, l)-ingenuity graph of order r(k, I) - 1 is included in some 
cliques with k - 1 vertices or in some independent set with 1 - 1 vertices. 
PROOF. Let G = (V,E) b e a (k,l)-ingenuity graph with r(k,l) - 1 vertices and let a be any 
vertex of G. 
Suppose each vertex subset with k - 1 vertices which includes vertex a is not a clique. 
We now consider graph G’ = (V’, E’) of order r(k, 1), whereV’=VU{b},E’=EU{byIayE 
E} u {ab}, b $ V. W e now prove G’ is an ingenuity graph. 
Let X be any set with k vertices of G. We consider the following three cases. 
CASE 1. a, b E X. Then /X - {b}l = k - 1, X - {b} C V, and X - {b} contains a. From the 
assumption, X - {b} is not a clique in G. So X - {b} is not a clique in G’, hence, X is not a 
clique in G’. 
CASE 2. b $ X. Then X C V, 1x1 = k. Since G is a (k,l)-ingenuity graph, X is not a clique 
in G. Hence, X is not a clique in G’. 
CASE 3. b E X and a $ X. Since a and b have the same neighborhood, this is similar to Case 2. 
X is not a clique in G’. 
By the above discussion, X is not a clique in G’. Since X is any set with k vertices, no clique 
exists with k vertices in G’. 
Let Y be any set with 1 vertices. We consider four cases. 
CASE A. a, b E Y. Because of ab E E’, Y is not an independent set in G’. 
CASE B. a E Y and b $ Y. Then Y E V, since G is a (k, I)-ingenuity graph, but IY 1 = 1, Y is 
not an independent set in G. Hence, Y is not an independent set in G’. 
CASE C. b E Y and a $ Y. Then, since a and b have the same neighborhood, this is the same 
as Case B, Y is not a clique in G’. 
CASE D. a $ Y and b $ Y. Then Y 2 V, since G is a (k,l)-ingenuity graph, but IYI = 1, Y is 
not an independent set in G. Hence, Y is not an independent set in G’. 
By the above discussion, Y is not an independent set in G’. Since Y is not any set with 1 
vertices, no independent set with 1 vertices exists in G’. 
Hence, G’ is a (k,l)-ingenuity graph. But there are r(k,l) vertices in G’. This contradicts 
Lemma 1.1, hence, we have proved that vertex a is contained in some clique with k - 1 vertices. 
Similarly, we can prove that vertex a is included in some independent set with 1 - 1 vertices. 
THEOREM 2.10. Let G be a (k, l)-ingenuity graph with r(k, 1) - 1 vertices, then there exist k - 1 
disjoint independent sets with 1 - 1 vertices in G, and 1 - 1 disjoint cliques with k - 1 vertices 
in G. 
PROOF. Assume G = G(V, E). 
(1) First, we prove that there exist k - 1 disjoint independent sets with 1 - 1 vertices. 
We now divide the vertex set V into disjoint independent subsets FI, Fz, . . , F,, S, 
where Fl,Fz,... , F, are independent sets with l- 1 vertices, and S is all of the remaining 
vertices not in Fl, Fz, . . . , Fm. Clearly, the independent set in S has at most I- 1 vertices. 
If there is no independent set with 1 - 1 vertices in G, let S = V, and m = 0. If there is 
at least one independent set with I- 1 vertices in G, the subsets Fl, Fz, . . . , F,,, and S are 
easy to get. We may allow m = 0 and S = 8. 
If m > k - 1, then we have the known conclusion. 
If m < k - 1, then we consider the graph G’ = (V’, E’) with r(k, I) vertices, where 
b $ V, V’ = Vu {b}, E’ = E u {ab I a 6 S}. 
As in the proof of Theorem 2.9, it follows that G’ is a (k, l)-ingenuity graph. But there 
are r( k, 1) vertices in G’, which contradicts Lemma 1.1. Hence, m > k - 1, that is, the 
conclusion is true. 
(2) Similar to (l), it follows that there are 1 - 1 disjoint cliques with k - 1 vertices in G. 
Some Properties of Ramsey Numbers 1193 
Finally, what we should state is that for every positive integer k > 2, according to monotonous- 
ness of the function, it is known that: 
(1) when k < 15, 2k/2 < (k - 1)2 + 1; 
(2) when k > 15, 2”i2 > (k - 1)2 + 1. . 
Hence, when 2 5 k 5 15, from the formula r(k, k) 2 (k - 1)2 + 1, we can get a better lower 
bound of the Ramsey number than from the formula r(k, k) 2 2k/2; when k > 15, the lower 
bound of Ramsey number by means of r(k, k) > 2k/2 is better. 
Similarly, when m = min{ k, 1) 5 15, we can get a better lower bound of Ramsey number by 
means of r(k, 1) 2 (k - 1)(1- 1) + 1 th an r(k, 1) 2 2k/2. When m = min{k, I} > 15, we can get a 
better lower bound of Ramsey number by means of r(k, 1) 1 2”i2 than r(k, 1) > (k - 1)(1- 1) + 1. 
Of course, if min{k, 1) is not big enough, but Ik - II is bigger, we can get a better lower bound of 
Ramsey number from r(k, 1) 2 (k - 1)(2 - 1) + 1. 
And what is more, in order to get a bigger lower bound of Ramsey number, we can divide k, 1 
into several integers by employing Theorem 2.1. 
For Ramsey number R( Rr , R2, . . . , R,), it is not difficult to get the same conclusion as in this 
paper, and we can deduce other bound formula of Ramsey number from the conclusions in this 
paper 
REFERENCES 
1. J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Macmillan, New York, (1976). 
2. H.L. Abbott, Lower bounds for some Ramsey numbers, Discrete Math. 2, 289-293, (1972). 
3. F.R.K. Chung and C.M.A. Crinstead, A survey of bound for classical Ramsey numbers, J. of Graph Theory 
7, 25-37, (1983). 
4. G. Exoo, A lower bound for r(5,5), J. of Graph Theory 13, 97-98, (1989). 


Some properties of Ramsey numbers 

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Some properties of Ramsey numbers

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