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 }> 2- UIUCDCS-R- 76-79^ 
 
 • A GENERALIZATION OF RAMSEY THEORY FOR GRAPHS 
 
 by 
 
 K. M. Chung 
 C . L . Liu 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
 W^JTYOF,u.,*oiS 
 ""WNA-CHAMPAIfiN 
 
+ 
 
 uiucdcs-r-76-79 1 * 
 
 A Generalization of Ramsey Theory for Graphs* 
 
 K. M. Chung 
 C. L. Liu 
 
 Department of Computer Science 
 
 University of Illinois at Urbana- Champaign 
 
 Urbana, Illinois 
 
 March 1976 
 
 This work was supported in part by the National Science Foundation under 
 grant MCS 73-O3I+O8. 
 
00.749-79? 
 
 1. Introduction 
 
 Let G,, Gp,...,G be graphs. The Ramsey number, R(G.., Gp, . . . ,G ), 
 is defined to be the least number p such that if the edges of the complete 
 
 graph K are colored in any fashion with c colors, then for same i the 
 
 th 
 subgraph whose edges are colored with the i color contains a G. [1, 2]. 
 
 When the G. ' s are complete graphs, this definition yields the classical 
 
 Ramsey numbers [3]« In this paper, we study a further generalization of 
 
 the classical Ramsey theory for graphs. 
 
 Let c and d be two integers such that c > d ^ 1. Let t denote 
 
 the binomial coefficient (.,). Given c distinct colors, we order the t 
 
 d 
 
 subsets of d colors in some arbitrary manner. Let G, , G p , . . ., G, be graphs. 
 
 The d-chromatic Ramsey number , denoted by R (G, ,G p , . . .,G ), is defined to 
 
 be the least number p such that if the edges of the complete graph K are 
 
 colored in any fashion with c colors, then for some i the subgraph whose 
 
 edges are colored with the i subset of colors contains a G. . In this 
 
 1 
 
 paper, we study the case c = 3> d = 2, and G,, G p , and G„ being complete 
 graphs. We shall denote the three colors OL, (3, 7, and order the three 
 subsets of two colors as {a,p}, (01,7), [$,7}. Thus, Rr (K. ,K.,Kj denotes 
 the smallest p such that if the edges of the complete graph K are colored 
 with the colors a, p, 7, there is either an a-p K. or an a-y K. or a 
 B-7 K, in the graph K . Without loss of generality, we shall assume 
 i ^j ^ i. 
 
2. Some Upper and Lower Bounds 
 
 We show in this section some upper and lower bounds on 
 R (K. ,K.,¥L„) . We derive first some upper bounds: 
 
 d. 1 J Jo 
 
 Theorem 2.1 : 
 
 E 3(K.,K.,^) S R(K.,K.) 
 
 Proof: If the edges of the complete graph K , p ^ R(K.,K.), are 
 colored with the colors a, p, 7 then there is either an cn-p K. or a 
 
 7 K. in K . □ 
 
 J P 
 
 Lemma 2.1 : Let N . N , N denote the number of a, p, 7 edges 
 
 incident with a vertex in K . A necessary condition that there is no a-p K. , 
 
 a-7 K., or p-7 K. in K is 
 3 l P 
 
 for every vertex in K . 
 
 P 
 
 o 
 
 Proof : Suppose N ' + N g Rx(K. ,,K.,Kj at a certain vertex A 
 ut p ■ c 1-1 j iJ 
 
 in K . Consider the complete subgraph G. whose vertices are those connected 
 
 to A by either en edges or p edges. There is either an cu-p K. ., or an 
 
 OL-y K., or a 8-7 K„ in G. . In the case that there is an CU-8 K. , in G., 
 2 1 k K l-l A' 
 
 such a K. , together with the vertex A will yield an Ct-p K. . 
 l-l K 1 
 
 The second and third conditions can be proved similarly. n 
 
Theorem 2.2 
 
 R^(K.,K ,K £ ) * ^{^(K.^K ,K £ ) + r|(K.,K ,K £ ) + R^K.^K^)} ] 
 
 Proof : Let 
 
 t = ^.i-S j-V + ^( K . )K >K ) + R 3 (K . ;K K ) 
 
 From Lemma 2.1a necessary condition that there is no a-B K., (X-y YL. f or 
 &-y K„ in K is 
 
 2(N a +N p +N 7 ) it-3, 
 
 or 
 
 N a + »e ♦ N 7 s |(t-3) 
 
 Therefore, if 
 
 N + N + N 1 f^(t-2)l 
 a B 7 ] 2 V ' ' 
 
 there will be either an a-R K., or an a-y K., or a 8-7 K„ in K . Therefore, 
 
 ^(K ± ,K,K ) * [|(t-2)l + 1= fit] 
 
 a 
 
 Theorem 2.3: If 
 
 R|(K i _ 1 ,K ,K^) + R^CK^K ,K^) Si p + 1 for p ^ i 
 
 then 
 
 e3(k. )K K ) S p 
 
Proof : 
 
 (l) Consider an arbitrary coloring of K . According to Lemma 2.1, 
 
 if there is no CC-fi K. or Oi-y K. in K , we must have 
 H 1 j p' 
 
 Also, 
 
 Thus 
 
 or 
 
 Given that 
 
 we obtain 
 
 N a + \ * ^'Wl'V " 1 (2 - 1) 
 
 >3 
 
 l^V^-i'Vr 1 ' (2 - 2) 
 
 N a + K p + M 7 = p " X • (2 - 3 > 
 
 ^'Vi'V^ 1 - u V p - ^ ( Wi'V 
 
 ^("i-i'V^' + ^Wi'V - p + x 
 
 E^(K i _ 1 ,K ,1^) + E|(K t ,K , ^K^) 5 p + 1 
 
 S 3(K.. 1 , K K^) ♦ e3(k., K K ) - p ♦ 1 
 
 (2) Adding (2.1) and (2.2), we obtain 
 
 2N + N + N ^ p - 1 (2.U) 
 
(2.3) and (2.1+) imply that N Q g 0. If N^ 
 is a p-7 K for I g p. 
 
 We derive now some lower bounds : 
 
 Theorem 2.k: 
 
 = at all vertices, then there 
 
 □ 
 
 3, 
 
 B£(K.,K ,X) >^(K._. f ,K ,K p ) + E|(K.„K K £1 ) + R^(K. t , K. „K^ f ) 
 
 + ^(K._ if ,K._. t ,K^ t ) -4 
 
 for any i' < i, j • < j, V < £. 
 
 Proof : The lower bound is obtained by the construction shown 
 in Figure 2. 1 
 
 ^(K...,^ V ) - 1 
 
 i." K j" K l-l'^ " 1 
 
 4( h „K j _ i „K i ,) - 1 ^(K.,.,,^.,,!^,) - 1 
 
 Figure 2.1 
 
where the circles represent complete graphs and the heavy lines between 
 
 graphs represents all the edges between the vertices in them. All the 
 
 edges represented by a heavy line are colored with the same color as shown. □ 
 
 As a special case, we have : 
 
 Corallary 2.U.1 : 
 
 p|(K .,K ,K J > k R^(K ,K ,K g ) - k 
 
 * ■ * £ M|J L#J L|j 
 
 Theorem 2.5 : 
 
 r|(k.,k^k^) > (i-l)(j-l) 
 
 Proof : Let p = (i-l) (j-l) . Let us identify the vertices of K 
 
 IT 
 
 by the ordered pairs (a,b), 1 ^ a ^ i-l and 1 ^ b ^ j-l. Consider the 
 following way of coloring K : An edge between two vertices (a,b) and (a,b r ) 
 is colored a. An edge between two vertices (a,b) and (a',b) is colored p. 
 All other edges are colored y. It can readily be checked that there is no 
 a-p K., or OL-y K., or p-7 K„ in K . D 
 
 3. Some Exact Values 
 
 For some small values of i, j, I, we are able to determine the 
 
 o 
 
 exact values of Rr(K.,K.,K ). 
 Theorem 3.1: 
 
 ^M' £ i' = j 
 
 Proof: Obvious. □ 
 
Lemma 3.1 : The only scheme to color the edges of K c with 2 colors 
 P 
 
 so that there is no monochromatic triangle is to have a circuit of length 5 
 of each color. 
 
 Proof : Let a and |3 denote the two colors used to color the edges 
 of K . If N is larger than or equal to 3 at a certain vertex, there is 
 either an a triangle or a p triangle. Thus, we consider the case N ^ 2. 
 Similarly, N£ 2. Since N ' + N = k, we must have N = N = 2 at all 
 
 6 U! p Cx p 
 
 vertices. It follows that the Oi edges form a circuit of length 5, and so 
 do the p" edges. □ 
 
 Theorem 3.2: 
 
 ^(K^K^Kj) 
 
 £ ^ 5 
 £ > 5 
 
 Proof: 
 
 3 
 
 (1) According to Theorem 2.5, Er(K ,K„,K. ) > k. 
 
 (2) According to Lemma 3.1* to avoid an cn-p" K~ in K , there must 
 
 be a 7 circuit of length 5; and to avoid an QL-y K_ in K,_, there must also 
 
 3 ? 
 
 be a p circuit of length 5. Consequently, these two circuits form a 
 
 0-7 \ 
 
 (3) Combining (l) and (2), we obtain r|(K-,K ,K ) = 5. 
 
 (U) The construction in (2) also shows that 
 e|(K 3 ,K 3 ,K ; ) > 5 for £ > 5 
 
 (5) According to Theorem 2.1 
 
8 
 
 (6) Combining (k) and (5), we obtain 
 
 I^(K ,K ,K ) = 6 f or I > 5 □ 
 
 Theorem 3.3: 
 
 I^(K 3 ,K v Kg) = i « 5, -6 
 
 4 = U 
 
 i = 5, 
 
 i i? 7 
 
 Proof: 
 
 ,3, 
 
 (1) The coloring scheme in Figure 3.1(a) shows that R p (K„,K, , Ki ) > 6. 
 
 (2) According to Theorem 2.2 
 
 p|(K 3 ,K v K^) ^ [^(K^K^) + ^(K^K^K^) + R^K^K^)} ] 
 
 = [1(^5+5)1 = 7 
 
 (3) Combining (l) and (2), we obtain r|(K ,K^,K^) = 7. 
 
 o 
 
 (U) The coloring scheme in Figure 3.1(b) shows that R (K„, K>,K ) > 7 
 and r|( k k^k^) > 7# 
 
 o 
 
 (5) We show now that PL (K~, K,,K^) ^ 8. Consider an arbitrary- 
 coloring of Ko with three colors. According to Lemma 2.1, N + N % 3. Thus, 
 
 8 x3 
 the total number of a and f3 edges is at most — ~- = 12. Since if there are 
 
 two or fewer a edges, there will be a p-7 K/-, we need only to consider the 
 
 case that there are 9 or fewer (3 edges. 
 
 Let 
 
 G denote the subgraph of Ko containing all (3 edges. If G 
 
 is a forest or if G contains only circuits of even lengths, according to 
 
 P 
 
 a well-known result that the vertices of G can be properly colored with 
 
> 
 
 (a) 
 
 (b) 
 
 (c) 
 
 solid lines are a edges 
 dotted lines are 3 edges 
 all other edges are y edges 
 
 Figure 3«1 
 
10 
 
 two colors, there are four vertices in G no two of them are adjacent. 
 
 P 
 
 These four vertices will form an a-y K. in Kq. 
 
 Suppose G contains a circuit of length 5 as shown in Figure 3.2(a). 
 P 
 
 Note that there is no additional 8 edges among the vertices v,, v , v„, v., v._ 
 
 1 d J 4 p 
 
 for otherwise there will be a 8 K~. Furthermore, two of the three vertices 
 
 Vr> v _, Vn must not be connected by a 8 edge either. Without loss of 
 
 generality, we assume that V/- and v are not connected by a 8 edge. 
 
 According to Lemma 2.1, N + N ^5- In other words N ^2. Since G„ 
 
 'ay 8 8 
 
 contains at most nine edges, there are at most two vertices of degree 3 
 
 in G . Thus, there are three vertices among v , v , v,, v., v that are not 
 
 connected to Vs and v . Furthermore, two of these three vertices are not 
 
 connected by a 8 edge. These two vertices together with v^ and v will form 
 
 an a- 7 K. in Ko. 
 
 Suppose G R contains a circuit of length 7 as shown in Figure 3.2(b). 
 
 If any two of the vertices among v ,v , ...,v are connected by additional 
 
 8 edges, there will either be a f] L in K Q or a circuit of length 5 in G Q . 
 
 j o p 
 
 We thus consider only the case that there is no additional 8 edges among 
 the vertices v, ,v p , . . .,v . Since there are at most nine 8 edges, Vn is 
 connected to at most two of the vertices v ,v_, ...,v by 8 edges. Suppose 
 Vn is adjacent to v . Either v q> v o> v ),> v V; or v «> v v Vc ;> V 7 w i^ form an 
 a-y Kv in Ko- 
 
 (6) Combining (h) and (5), we obtain r|(K ,K^,K ) = b|(K ,K^,K 6 ) = 8. 
 
 o 
 
 (7) The coloring scheme in Figure 3.1(c) shows that R^(K , K, , K ) > 8. 
 
 (8) By Theorem 2.1, r|(K ,K^,K ) < R(K ,YL^) = 9. Therefore, 
 r|(K 3 ,K v K^) = 9 for I > 7. D 
 
11 
 
 Theorem 3,k: 
 
 
 
 
 f 9 
 
 
 ' 
 
 11 
 
 r|(k 3 ,k 5 ,k £ ) = 
 
 
 12 
 
 13 
 I 11+ 
 
 for I = 5 
 
 £ = 6 
 
 1 = 7, 8 
 
 ^ = 9, 10, 11, 12, 13 
 
 I § 1*1 
 
 . Y 
 
 6 
 
 V 
 
 7 
 
 v 8 
 
 • V; 
 
 (a) 
 
 00 
 
 Figure 3-2 
 
12 
 
 Proof : 
 
 (1) According to Theorem 2.5, r|(K„,K ,K ) > 8. 
 
 (2) To show that Rr(K , K , K ) ^ 9j we consider an arbitrary- 
 coloring of K Q . Let G denote the subgraph of Kn containing all OL edges and 
 P edges, respectively. We examine the following cases: 
 
 (i) Suppose G is either a forest or a graph all of whose circuits 
 are of even lengths. Since the vertices of G can be properly colored with 
 two colors, there are five or more vertices in G no two of which are 
 adjacent. Thus, there is a p-7 K,_ in K Q . A similar argument can be applied 
 
 to G . 
 P 
 
 (ii) Suppose that G contains a circuit of length 5. If no two 
 
 of the five vertices in the circuit are connected by a p - edge, then there is 
 
 a a- 7 K in K Q . If any two of the vertices are connected by a (3 edge, then 
 
 there is a OL-P> K in K n . A similar argument can be applied to G . 
 3 y P 
 
 (iii) Suppose that G contains a circuit of length 7> which we 
 
 denote C . As illustrated in Figure 3.3> no vertex in C can be connected 
 
 to two or more of the vertices in C by p edges. It follows that there can 
 
 be no circuit of length 7 or 9 in G since such a circuit in G must contain 
 
 P P 
 
 5 or more of the vertices in C . Consequently, the arguments in (i) and (ii) 
 can be applied to G . 
 
 (iv) Finally, suppose that G contains a circuit of length 9- Let 
 the vertices of the circuit be labeled v,,v p , ...,v as shown in Figure 3. Ma) 
 Figure 3. Mb) shows exhaustively that no ordering of the vertices can yield 
 a circuit of length 9 in G without forming an a-{3 K„ in K , where a path 
 in the tree representing a possible ordering of the vertices. 
 
Figure 3-3 
 
 13 
 
 V- 
 
 A path in the tree represents a possible (3 path in 
 the a 9-gon, and is terminated when exhausted or when 
 an a-3 triangle is formed. 
 
 Figure ^> .k 
 
Ik 
 
 (3) Combining (l) and (2), we obtain R^(K,.,K ,K ) = 9- 
 (h) The coloring scheme in Figure 3.5(a) shows that 
 
 I^(K 3 ,K 5 ,K 6 ) > 10. 
 
 (5) According to Theorem 2.2, e|(K ,K ,K.) < 11. 
 
 (6) Combining (k) and (5), we obtain B~(K ,K ,K£) = 11. 
 
 (7) The coloring scheme in Figure 3.5(b) shows that 
 
 b|(k 3 ,k 5 ,k 7 ) > 11. 
 
 (8) To show that Rr(K~,K ,Kn) ^ 12, let us consider an arbitrary 
 coloring of K-. Q . Again, let G and G denote the subgraph containing all CL 
 edges and p edges, respectively. According to Lemma 2.1, at any vertex 
 
 N + N % k 
 
 a p 
 
 N + N ^ 8 
 a 7 
 
 Since 
 
 W + N + N =11 
 a p 7 
 
 it follows that at any vertex N =3 or k. 
 
 P 
 
 (i) Suppose that there is a vertex of degree k in G . As 
 
 P 
 
 Figure 3.6 shows, since the value of N for the vertices v ? , v~, v. , v is 
 at least 3, there are at least eight (3 edges connecting the four vertices 
 
 Y 2' v 3' v k' v 5 to the seven vertices Vg, v , v g , v^, v 1Q , v^, v^. 
 
 Consequently, two of the vertices in v n , v„, v. , v_ must be connected to 
 
 one of the vertices in v,-, v , Vq, v , v , v , v „ by p edges. We conclude 
 
 that there is a circuit of length h in G . Let v.., v , v., v^ denote the 
 
 & P 1 2' 3' 6 
 
 vertices in this circuit as shown in Figure 3.7(a). Suppose each of v, 
 
 and v^ is adjacent only to one of the vertices v. , v , v , Vn, v , v in , v , v p , 
 
 There are six vertices among v,, v , v , Vn, v , v, n , v ,, v, p that are not 
 
15 
 
 /[ 
 
 K 
 
 r i 
 
 / i 
 
 / i 
 
 / i 
 
 / i 
 
 \ \ 
 
 1 \ 
 
 \ \ 
 
 > * 
 
 \ ; 
 
 t 
 1 / 
 1 / 
 , / 
 
 V! 
 
 1 A 
 i / 
 
 N- 
 
 is 
 
 (a) 
 
 (b) 
 
 (c) 
 
 solid lines are a edges 
 dotted lines are p edges 
 all other lines are y edges 
 
 Figure 3«5 
 
v 2/f' v 3/ V/t v 5 % > 
 
 / \ / \ / \ / 
 
 v 6 V T v 8 v 9 v io v n V 12 
 
 Figure 3*6 
 
 16 
 
 v 
 
 JV-, 
 
 V 
 
 6r- t v l 
 i ' 
 
 3 
 
 ■V 
 
 #v 8 
 
 • V 
 
 • V 
 
 V 
 
 • V 
 
 9 
 10 
 
 11 
 12 
 
 v 8 V 7 V 5 % 
 
 \ n i 
 
 \ / \ ' 
 
 \ / \ / 
 
 v 6*— 1 V 1 
 
 V, 
 
 v 3^"A-2 
 
 / » 
 
 / \ 
 
 V 10 v 9 
 
 v 
 
 V 
 
 11 
 
 12 
 
 (a) 
 
 00 
 
 Figure 3-7 
 
IT 
 
 connected to v and v_ by B edges. Since R(3,3) = 6 either there is a 
 a-p K~ among these six vertices or there is a 7 L among these six vertices. 
 The three vertices that form a 7L together with v, and v~ will form a 
 QL-y K in K no . A similar argument can be applied to the case where each of 
 v ? and v,- is adjacent only to one of the vertices among v. , v _, v , Vg, v~, 
 
 v io' v n' v i2- 
 
 We now consider the situation in Figure 3.7(b). 
 
 Consider the 5 vertices v,,v ,v ,v , v . According to Lemma 3«1> 
 they must form a p pentagon, or else there is an a- 7 K,, which, together with 
 the vertices v p and v> will form a a- 7 Kj.. Similarly, the vertices v ,vn, 
 v n ,v ,v must also form a P-pentagon. But since the value of v Q at the 
 
 y 11 Ld p 
 
 vertices v. ,Vj_,v ,Vg,v ,v _ is at most 3, there is no p edges from any one of 
 v. ,v ,v to any one of v ,Vn,v . Hence the vertices v-,v, ,v,_,v ,Vg form an 
 0-7 K_ in K.p. 
 
 (ii) Suppose N = 3 at all vertices. As illustrated in Figure 3.8(a), 
 P 
 
 if any two of v p , v~, v. are connected to one of v , v,-, v , Vg, v q , v- , v.-, 
 
 v, by a B edge there is a circuit of length h in G . On the other hand, 
 id p 
 
 as illustrated in Figure 3.8(b), if no two of v ? , v_, v. are connected to one 
 
 of v , v^-, v , Vg, v , v , v _., v-.p by a p edge, there is either a circuit 
 
 of length h or a circuit of length 6 in G . 
 
 P 
 
 If there is a circuit of length K in G as shown in Figure 3.9(a), 
 
 P 
 
 at most two of v_, v r , . . .,v in , v no are connected to v, and v, by p edges. 
 5 O 11 ±d 14 
 
 Suppose v , V/-, v , Vq, v , v are six vertices that are not connected to 
 v. and v, by p edges. Since R(K_,K_) = 6, either there is a p K. or an 
 OL-y K_ among them. The three vertices that form an OL-y K_ together with 
 v. and Vi will form an OL-y K in K,p. 
 
18 
 
 v i 
 / \ 
 
 v 5 v 6 v 7 v 8 v 9 v l0 v ll v i2 
 
 v, 
 
 \ 
 
 / ! \ 
 
 / \ h \ 
 i \ \ \ 
 
 I \ i \ t \ 
 
 v 5 v 6 V T v 8 v 9 V 10 
 
 (a) 
 
 v ll ^12 
 
 (b) 
 
 Figure 3 '8 
 
 i i 
 i i 
 i 1 
 
 V- 
 
 v, 
 
 5 
 
 * v 8 
 • v 
 
 • v 
 
 V 
 
 • V 
 
 9 
 10 
 
 11 
 
 12 
 
 r 6< 
 
 v l 
 r- 
 
 v 
 
 Vr 
 
 > V 5 
 
 / 
 / 
 
 • v, 
 
 • V 
 
 • V 
 
 • V 
 
 • V 
 
 • V 
 
 7 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 (a) 
 
 (b) 
 
 Figure 3.9 
 
 If there is a circuit of length 6 in G as shown in Figure 3.9(b), 
 
 P 
 
 three of the vertices among v, , v p , v , v, , v , v^ will form 7 K , 
 
 Without 
 
 loss of generality, we can assume that the vertices v,, v~, v form a 7 K . 
 
 Among the vertices v , Vn, v q , v,_, y__, v, ? only three of them are 
 
 connected to the vertices v, , v , v by 8 edges, since N ^ 3 at each 
 
 13 5 P 
 
 vertex. Assume that the vertices v , Vq, v are not connected by v , v , v 
 
19 
 
 by p edges. If v , v ft , v are all connected by p edges, they form a £ K~. 
 
 Otherwise, two of them that are not connected by a p edge together with 
 
 v,, v_, v_ form an OL-y K c . 
 J- o 5 ? 
 
 (9) Combining (7) and (8), we obtain r|(K ,K , K_ ) = r|(K ,K ,Kq) = 12, 
 (10) The coloring scheme in Figure 3.5(c) shows that 
 r|(K 3 ,K 5 ,K 9 ) > 12. 
 
 (11) To show that 
 
 e|(K ,K ,Kg) ^ 13 for i ^ 13 
 we note that Rr(Kp,K ,Kj = 5 and Rr(K„,K> ,K ) = 9. According to Theorem 2.3 
 ^(K^K ,Kg) + i^(K 3 ,K v K^) =13 + 1 for i ^ 13 
 
 it follows that 
 
 Also, 
 
 e|(K 3 ,K 5 ,K^) =g 13 for I < 13 
 
 (12) Combining (10) and (ll), we obtain 
 
 e|(K 3 ,K 5 ,K^) = 13 f or 9 < i ^ 13 
 
 (13) According to Theorem 2.1 
 
 r|(K 3 ,K 5 ,K^) g R(K 3 ,K 5 ) = Ik . 
 
 e|(K ,K ,K ) > 13 for I > 13 
 
 *3' 5' i 
 
 Therefore, we obtain 
 
 r3(K ,K ,Kg) = 11+ for i > Ik a 
 
20 
 
 Theorem 3.5: 
 
 B^WV = 10 
 
 Proof: 
 
 (1) According to Theorem 2.5, r|(K, ,K, ,K, ) > 9. 
 
 (2) To show that Rp(K. ,K. ,K. ) ^ 10, we consider an arbitrary 
 coloring of K ir . Again, let G and G denote the subgraph containing all 
 a edges and (3 edges, respectively. According to Lemma 2.1, at each vertex 
 
 N + N ^ 6 
 
 N + N =g 6 
 
 N Q + N ^6 
 P 7 
 
 It follows that N = N = N =3 because N + N + N = 9. Since 
 R(K , K, ) = 9, there must be an OL K or else there is a p-7 K. . Let the 
 vertices of the a K be labeled v, , v p , v~ as shown in Figure 3.10. No two 
 of these three vertices can be connected to the same vertex in G since 
 that will yield either an CH-p K, or OL-y K. in K, n . We thus have the 
 configuration shown in Figure 3.10. Let us now exhaust the four cases in 
 which there are three, two, one, and zero a edges among the vertices 
 
 v v V V 
 
 (i) Suppose there are three edges among the vertices v. , v , v^- 
 as shown in Figure 3.11(a). In this case, there is no a edges between the 
 vertices in v, , v„, v , v, , v , v.- and the vertices in v , Vq, v , v . 
 Among the four vertices v , Vq, v , v there must be two that are not 
 connected by an a edge, otherwise v , Vn, v , v „ will form an CC K, . Suppose 
 these two vertices are v and Vn. Clearly, v., v. , v , Vo form a p-7 Kk. 
 
21 
 
 • • • 
 
 V„ Vn V V 
 
 7 8 9 10 
 
 Figure 3-10 
 
 (ii) Suppose there are two a edges among the vertices v. , v , v^ 
 as illustrated in Figure 3.11(b). Among the vertices v , Vn, v Q , v _, there 
 are at most two that are connected to the vertices v, and V/- by Q! edges 
 because H ' = 3 at every vertex. Suppose v is not connected to v, or v^ 
 by an oc edge. The vertices v , v, , v.-, v form a p-y K, . 
 
 (iii) Suppose there is one cc edge among the vertices v. , v , v^ 
 as illustrated in Figure 3.11(c). Since v^- is connected to two of the 
 vertices among v , Vq, v , v , if v,- is not connected to v and Vn, then 
 
 v r v 6' V V Q forma P"^ K v 
 
 ■ (iv) Suppose there is no a edge connecting the vertices v., v , v^ 
 as illustrated in Figure 3.11(d). Among the vertices in v , Vn, v q , v,_, 
 there is one that is connected to at most one of the vertices in v. , v , v^- 
 by an a edge because there is a total of six a edges connecting the vertices 
 in v. ,v ,v^- and the vertices in v ,Vo,v ,v . Suppose v is connected to 
 v. only as illustrated in Figure 3.11(d). Clearly, v,,v ,V/-,v form a P-7 
 
 K, 
 
 (3) Combining (l) and (2) we conclude that R^(K. ,K. ,K- ) =10. D 
 
22 
 
 00 • ■ 
 
 V T v 8 v 9 v io 
 
 V T v 8 v 9 v io 
 
 (a) 
 
 00 
 
 O 
 
 v 7 v 8 v 9 v io 
 
 • a • 
 
 (c) 
 
 V T v 8 v 9 V 10 
 (d) 
 
 Figure 3 • 11 
 
23 
 
 k. Concluding Remarks 
 
 There are many possible extensions to the problem we have studied. 
 For example, it will be interesting to study the bichromatic numbers 
 
 o 
 
 Br(G,,G p ,G _) where G , Gp, G~ are not complete graphs. Some partial results 
 
 o 
 
 on the values of Br (P., P., P.) have been obtained [k], where P. denotes a 
 
 <— 1 J Z X 
 
 path of length i. Also, one might wish to study the values of B, (G , G , . . ,,G. ) 
 
 CL _L cL "G 
 
 for other values of c and d. 
 
2k 
 
 References 
 
 [1] Burr, S. A., Generalized Ramsey Theory for Graphs - A Survey, 
 to appear. 
 
 [2] Chvatal, V. and F. Harary, Generalized Ramsey Theory for Graphs, 
 Bull. Amer. Math. Soc. 78 (1972), h2^-k2G. 
 
 [3] Ramsey, F. P., On a Problem of Formal Logic, Proc. London Math. 
 Soc. 30 (1930), 26I1-286. 
 
 [k] Lee, D. T., Private Communication. 
 
LIOGRAPHIC DATA 
 ET 
 
 1. Report No. 
 
 U IUC DC S-R- 76-79I+ 
 
 3. Recipient's Accession No. 
 
 itle .uiJ Subt it le 
 
 A Generalization of Ramsey Theory for Graphs 
 
 5. Report Date 
 
 March 1976 
 
 6. 
 
 uthor(s) 
 
 K. M. Chung and C. L. Liu 
 
 8. Performing Organization Rept. 
 No. 
 
 crtorming Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urbana- Champaign 
 
 Urbana, Illinois 618OI 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 MCS-73-03^08 
 
 Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 Supplementary Notes 
 
 Abstracts 
 
 Let c and d be two integers such that c > d 1 1. Let t denote the binomial 
 efficient ( ). Given c distinct colors, we order the t subsets of d colors in some 
 bitrary manner. Let G-,G , ...,G, be graphs. The d-chromatic Ramsey number , denoted 
 R d (G ,G , . . .,G ), is defined to be the least number p such that if the edges of the 
 mplete graph K are colored in any fashion with c colors, then for some i the 
 
 bgraph whose edges are colored with the i subset of colors contains a G. . In this 
 
 1 
 
 port, we study the case c = 3, d = 2, and G , Q , and G, being complete graphs. 
 
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