On Ramsey Numbers Defined by Factorizations of Regular Complete Multi-Partite Graphs

Transcription

1 Western Michigan University ScholarWors at WMU Dissertations Graduate College On Ramsey Numbers Defined by Factorizations of Regular Complete Multi-Partite Graphs James M. Benedict Western Michigan University Follow this and additional wors at: Part of the Mathematics Commons Recommended Citation Benedict, James M., "On Ramsey Numbers Defined by Factorizations of Regular Complete Multi-Partite Graphs" (1976). Dissertations This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWors at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWors at WMU. For more information, please contact

2 ON RAMSEY NUMBERS DEFINED BY FACTORIZATIONS OF REGULAR COMPLETE MULTI-PARTITE GRAPHS by James M. Benedict A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the Degree of Doctor of Philosophy Western Michigan University Kalamazoo, Michigan August 1976

3 ACKNOWLEDGEMENTS I wish to than all my former professors of Central Michigan University and Western Michigan University for their patience and understanding while teaching me mathematics. I wish to than the members of my committee for woring so efficiently under the pressure of a dwindling amount of available time. I wish to than Miss Marie Smith for exclusively devoting a wee of her time to typing this manuscript. I particularily wish to than Professor Gary Chartrand of Western Michigan University, Professor William Laey of Central Michigan University, and Mrs. Helen Spotts of Jonesville High School for providing guidance at just the right time in just the right manner. James M. Benedict ii

4 For Rolla Ann who wored harder than I. For Nathan and Laura, who missed their Daddy, and never did understand why. iii

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6 76-28,432 BENEDICT, Janies M., ON RAMSEY NUMBERS DEFINED BY FACTORIZATIONS OF REGULAR COMPLETE MULTI-PARTITE GRAPHS. Western Michigan University, Ph.D., 1976 Mathematics Xerox University Microfilms, Ann Arbor, Michigan 48106

7 TABLE OF CONTENTS CHAPTER PAGE I PRELIMINARIES... 1 An Overview of Ramsey Numbers General Ramsey Numbers... 7 The i-th Ramsey N u m b e r s...14 Closing Comments II ON SUBGRAPH RAMSEY NUMBERS FOR THE REGULAR COMPLETE MULTI-PARTITE GRAPHS 22 Lower B o u n d s Exact R e s u l t s III MOST SMALL i-th RAMSEY N U M B E R S Preliminary Results Computations IV ON TWO VERTEX PARTITION i-th RAMSEY NUMBERS 75 A Review of the Lexicographic P r o d u c t The i-th Chromatic Ramsey Numbers.. 78 The Evaluation of a(i,) On i-th Vertex Arboricity Ramsey N u m b e r s iv

8 CHAPTER I PRELIMINARIES The central purpose of this dissertation is to investigate certain generalizations of existing ramsey numbers. The focus of Chapter I is to provide a unifying framewor within which all previous ramsey numbers occur as special cases. Arguments are given to support the contention that only those numbers which occur within this framewor should be called ramsey numbers. In the final three chapters of this dissertation a particular restriction of the general case is studied. New ramsey numbers are discovered all of which are extensions of currently nown ramsey numbers. A frequent procedure of generalization in graph theory is the extension of theorems involving the complete graphs to theorems involving a larger family of complete multi-partite graphs. Many examples of this type of extension from topological graph theory are found in White [ 36]. The target of such extensions is often the octahedral graphs [20] or arbitrary regular complete multi-partite graphs [36], Similar results from factorization theory are contained in [24,25]. This trend is followed in the final three chapters of this dissertation since the notion of the complete 1

9 graph in ramsey theory is extended to all regular complete multi-partite graphs. Special attention is given to the octahedral graphs. The next section provides an overview of ramsey theory. The standard graph theoretical terms and notation (as found in Behzad and Chartrand [ 1 ] or white [36]) are used. It is simply noted here that graphs are the usual (single-edged, finite, loopless, undirected) combinatorial entities built upon vertices and edges. Section 1.1 An Overview of Ramsey Numbers A brief overview of ramsey numbers is given in this section. A more thorough treatment may be found in the excellent surveys b y Burr [5] and Harary [21]. Even today these surveys nearly represent the state of the art. In 1930 [31] Ramsey proved (in a non-graphical context) that if G is an infinite graph having a countably infinite vertex set, then either G or G contains the complete infinite graph of order as a subgraph. Moreover, for positive integers m and n there is a smallest positive integer p such that given any graph G of order p either is a subgraph of G or K n is a subgraph of G. Determining these numbers p for various m and n is the classical ramsey problem. The notation and concepts have advanced as follows.

10 3 A graph G is said to have a factorization into the (al) subgraphs (factors) f ^ F 2»..., (written G = F- F 0... F. or more compactly G = F.) if 1 2 j=1 j (i) V (F.) = V(G) for 1 s j S, (ii) U E(F.) = E (G), 3 j=l 3 (iii) E ( FS ) 0 E (Ft ) = <6 whenever s ^ t. A semipartition of a set S is a collection of sets S., S~,...# S, having the properties (i) (J S. = S 1 2 j=1 3 and (ii) Sj fl St = <4 whenever j ^ t. It may be said then that a factorization of a graph G is a finite collection of spanning subgraphs whose edge sets form a semi-partition of E ( G ). If the graph F is a subgraph of the graph G we write F c G. For positive integers n^, n 2» n^ the ramsey number r (n^,n2 #...,n^) is the least positive integer p such that whenever Kp = F^ then Kn c F^ for at least one j with 1 j. Thus the classical ramsey problem (in two factors) has been to determine r(m,n) for every two positive integers m and n. The degree of difficulty of the classical ramsey problem is reflected in the fact that only seven non-trivial such numbers are nown (see Figure 1.1). The difficulties of the classical ramsey problem probably account for the growth of the scope of the

11 Classical Ramsey Numbers r (3, 3) = 6 r (3, 3, 3) = 17 r (3,4) = 9 r (3,6) = r(4,4) = 18 r (3, 5) = 14 r (3,7) = 23 Figure 1.1 problem. Starting in 1967 [19], the focus of the problem has changed to include the generalized ramsey number. For the (si) graphs G^, G 2,..., G^ the generalized ramsey number r(g^, Gj,..., Gj^) is the least positive integer p such that whenever K = 0 F. then there is p j=l 3 some j with 1 s j for which Gj c F ^. That r(g^, G 2 #...» G^) exists for any (si) graphs Gl' G 2' ***' ^ directly from Ramsey's original wor. The generalized ramsey problem is to find r (G l# G 2»... G^) for every graphs G^, G 2,..., G^ (or at least the standard interesting g r aphs). As seen in Burr's survey [ 5 I much has been accomplished along these lines. Since 1974, evolutions from the generalized ramsey numbers have occurred in three differing varieties: (i) properties of factors of Kp other than graph theoretic inclusion; (ii) factors of graphs other than Kp ; and (iii) restricted factorizations of K have been P studied. Examples of (i), (ii), and (iii) are presented.

12 Chartrand and Polimeni [8 ] define x(n ^* n 2 '» for the (si) positive integers n^, n 2,» to be the least positive integer p such that whenever K = F. $ then there is some j with 1 j a. for P j=i ^ which x(f j) 25 n j (where x(g ) denotes the chromatic number of the graph G ). Further generalizations are given in [3,26,27,28]. Beinee and Schwen [2 ] define for the positive integers m and n to be the least positive integer p such that whenever K(p,p) = F 1 F 2 it follows that K(m,n) c F^ or K(m,n) <z "S2 Similar generalizations are given in [17,18]. As a concluding example, Sumner [33] defines rc (G^,G2 > for the graphs G^ and G 2 to be the least positive integer p such that whenever = F^ F 2, where both F 1 and F 2 are connected graphs, it follows that G 1 c F x or G2 c F 2. It is the purpose of the next section to define a general ramsey number and to develop some suggestive and unifying notation. It is hoped that both the definition and notation are broad enough to contain the mainstream ramsey numbers as special cases, yet narrow enough to be meaningful. Before this can be accomplished, it must be nown when a given number (which arises as a result of a factorization of a graph) is to be considered a ramsey number. Graham has been quoted (see [22]) as

13 6 regarding "any criterion for partitioning the lines of a graph as a ramsey-type problem." The author disagrees with this assessment. Every ramsey number to date has been discovered with the aid of a certain proof technique, an example of which now follows. To show that r(3,3,3) = 17, it is demonstrated 3 that whenever = F. then K, c F. 17 j=1 J 3 3 for some j (1 s j 3). It follows that r(3,3,3) 17. To see that r(3,3,3) i 17 it suffices to display a factor- 3 ization K, c = F. 16 j=1 3 with K, 3 F. for every 3 j (1 j 3). The main point is this: since K 3 c F* c F implies K 3 c F and since p s implies Kp c: Ks then for every 1 s p 16 there is a factorization 3 K = F. with K_ gf F. for every j (1 j 3). p j * l ] Moreover, for every p a 17 and every factorization 3 Kp =. Fj there is K 3 c Fj for some j (1 * j 3). This phenomenon can be thought of as "continuity in the integers". Every nown ramsey number has implicitly used this phenomenon within its proof. It seems reasonable to require this continuity in the integers for a factorization problem to be classified as a ramsey problem. Care will be taen in the general definition to see that the desired "continuity" is preserved.

14 7 Section 1.2 General Ramsey Numbers The purpose of this section is to define the general ramsey number for graphs. Some attention is paid to the three main components of ramsey number definitions which are: (i) a family of graphs is being factored; (ii) certain specialized properties (of a type described below) of each factor may be required; and (iii) arbitrary restrictions may be placed on the entire factorization. Attention is primarily directed toward items (i) and (ii). It is suggested that ramsey numbers which arise from restricted factorizations be called restricted ramsey numbers. This will allow us to thin of the general ramsey number as an unrestricted ramsey number. Sumner's wor [33] is the only nown example of restricted ramsey number theory to date. The types of families of graphs and properties of factors to be considered are now specified. The indexed family of graphs = (Gp jp Z+ ] is called an ascending family of graphs if p s s implies Gp c G s. For example tkp P z + ) is 311 ascending family whereas { p Z+ ) (the family of cycles) is not. Only ascending families will be considered for the general ramsey number.

15 8 Now let rt be a property of graphs such that every given graph G either has or fails to have property.n. For example, tt could be the property of having chromatic number at least five. We see that has property tt while K 4 does not. The property tt is called an ascending property if whenever the graph G has property tt and G c H then it follows that the graph H has property tt. It may be noted that the above example is an ascending property while "having chromatic number equal to five" is not. The general ramsey number is now defined. Definition 1.1 Let $ = (Gp p Z+ } be an ascending family of graphs, and let tt^, tt2 *» Tr be (si) ascending properties of graphs. Define the general ramsey number tt2' '*' n^ to be the least inteser P (if it exists) such that whenever Gp = Fj then Fj has property tt j for some j (1 j s ). Some notation is developed after which it is shown that the desired continuity in the integers is present in the general ramsey number. For graphs G and H with V (G) 0 V (H) i* <b we define the graph G PI H by (i) V (G n H) = V(G) 0 V (H) and (ii) E (G 0 H) = E (G) 0 E (H).

16 9 Note that if V (G) fl V(H) / $ and if G = F. then j=l 3 g n h = (f. n h ). j=i 3 Again for the graphs G and H, we can always insure that V(G) fl V(H) = 0 b y relabeling the vertices of one of the graphs if necessary. With this convention in mind, we define G U H only for graphs with disjoint vertex sets b y (i) V (G U H) = V(G) U V(H) and (ii) E (G U H) = E(G) U E(H). For the positive integer n, we write ng to denot n the graph U G = G U G U... U G (n copies of G ). We j=l often write G ^ to denote the j-th copy of the graph G. That is, G = G ^ up to a possible relabeling of the vertices. Using these ideas, we can describe that Cg is constructed from two 1-factors each of order 6 2 by the notation = (3K-) 6 j=l 2 Theorem 1.1 (continuity in Z+ ) Let $ = { G j j 6 Z+ } be an ascending family of X 1 graphs and let tt2 #...» be (2:1 ) ascending properties of graphs. Let p be an integer such that whenever G = F. P j=i 3 then F j has property tt j for some j with 1 < j.

17 10 Let b be an integer such that there exists a factoriza- tion G. = H. for which H. fails to have property j=l 3 3 tt j for every j with 1 j. Then for every n i p it holds that whenever G = P. there is an integer j with 1 j n j=l 3 for which Fj has property ffj. Also, for every n b there exists a factorization G = F. where F.fails n j=l 3 3 to have property Kj for every j with 1 j.. Proof: Let n i p and let Gn = Fj. Since Gp c Gn 3 we have G = (F. fl G ). By the hypothesis on p we P j=i J P can assume F^ fl Gp has property tt.^. Since F1 H G^ c F^ and since is an ascending property then F^ has property. N o w let n b. Since Gn c Gj^ we have the fac- torization G = S (H, (1 G ). Since (i) H. does not j=l 3 3 have property rtj (1 j ), (ii) TTj is an ascending property (1 j ), and (iii) Hj fl Gn e Hj (1 j ) then Hj fl Gn cannot have property rij for each j with 1 j The importance of Theorem 1.1 is that we are guaranteed that b+1 r ^ tt2< tt^) p.

18 11 It is easy to concoct examples of numbers defined in a manner similar to the general ramsey number which fail to be continuous in the integers since either the family of graphs or the properties are not ascending. As an example of the former case, tae $ = (Gp Gp = Kp if p ^ 10} U (G10), where G 10 * 101^. Further, let rr^ be the property of having three or more edges for j = 1 and for j = 2. Then r $ (n^, rr2 ) = 4 even though G 1Q = 101^ 10K1 is a factorization in which both factors fail to have three or more edges. As an example of the latter case, let (r = CKp p Z+ } and let tt be the property of having order equal to three. Then r^ (tt, tt) = 3 even though K4 = P4 P4 where both factors fail to have property tt. More theorems basic to "general ramsey theory" are now presented. Theorem 1.2 (symmetry) Let $ be an ascending family of graphs, let tt^* tt2 / tt^ be (si) ascending properties of graphs, and let r^ (tt^, tt2, # tt^) = p. Then for any permutation t of (1, 2,...» }, it follows that r,(nt(x), "t(2)... n T ()) = p. Proof: Let t = (Gfcjt z+ ) be an ascending family of graphs. To show that r $(nt (i)' nt(2)' ' % () ^ * p '

19 let g = F. be a factorization of G^ and let P j=i 3 P CT -1 = T Consider G = F,.A. P j=1 a n ; By the definition of p there exists jq w ith U jq < for which the graph F^ ^ j has property tt^. Let j = a (jq ) Then rr. = tt, = tt This implies that F. has pro- 3o cr- 1 <3) T(:l) 3 perty ttt j j j which establishes an upper bound for r (nt(l>' nr(2)...nt ()) * The desired lower bound is similarily established Theorems 1.1 and 1.2 are generalizations of corresponding results from previously existing ramsey theory as is Theorem 1.3, which needs the following definition For the ascending properties tt and of graphs we write t * tt given tt) if ir is true for a given graph G whenever tt is true for that graph G. For example, y (G) i 3 «- "x (G) a 4". Theorem 1.3 (monotonicity) Let i be an ascending family of graphs, let rxy tt2,..., tt^ and " " ^ be two P airs of (ssl) ascending properties of graphs, let r$^l' n2' *'*' n^ =P' and assume * for I s j <. Then r ^ f ^, <]r2,..., # ) p.

20 Proof: We specify $ = {G. It Z+ } and assume G = F.. t P j=l 3 There is some j for which the graph Fj has property follows The factorization is arbitrary so the conclusion Theorem 1.3 maes a statement concerning the general ramsey number for a fixed family and changing properties. The situation is reversed in Theorem 1.4 which needs the following definition. For the ascending families $ = C<3p P Z+ } and ^ = (Hp p Z+ } of graphs we write -* * if G c H for each p Z+. Theorem 1.4 has no p p analogous result in ordinary ramsey theory. Theorem 1.4 (anti-monotonicity) Let and be ascending families of graphs for which $ -» \ r, let tt^, tt2, / tt^ be ascending properties of graphs, and let r $ 'tt2,» tt^) = P * Then ^ ( t^, tt2,..., tt^) p. Proof: Let i [Gt t Z+ }, let f = {Ht t. Z+ ), F.. Since G c H it follows that 3 P P j = l GP (Gp n V = ^ * 3 -n Gp>- However, b y the definition of p there exists some j for which F^ fl Gp has property tt j (1 j ).

21 Since T7j is an ascending property and since Fj fl Gp a Fj, then Fj has property ttj. This implies the validity of the theorem We end Section 1.2 with a final observation concerning general ramsey numbers. Given and rr^» tt2,..., tt^ as usual, let TT+1 be the ascending property "the edge set is non-empty". Then r $ (TTl' Tr2 V r * (TTl' " 2... "' TT + l ) Section 1.3 The i-th Ramsey Numbers We narrow the discussion to examine general ramsey numbers with respect to a certain type of ascending family. A graph G having vertex set V(G) is called p-partite if the vertex set can. be partitioned into p (il) partite sets V 2,..., Vp so that each partite set is an independent set of vertices of G. Note that every graph G can be thought of as p-partite for each integer p such that s P s V(G). A graph K is called complete p-partite if K is a p-partite graph such that for each two vertices u and v of K we have uv e E(K) if and only if u and v belong to distinct partite sets. Given that Vjl = n j for

22 j s p we write K = K ( n 1> n 2,...» np ). Each such K is unique up to a reordering of partite sets. If the number of partite sets of K is to be left unspecified, we refer to K as a complete multi-partite graph. As examples, note that Kn = nk.^ = K(n) for n 6 Z+, and the star graphs are K(l,n) for n Z+. If each partite set of the complete p-partite graph K O ^, n 2,..., np ) has exactly i (2:1) vertices we write K p(i) rather than K(i, i,...» i) to denote this graph. Note that Kp(i) = Kp for eac^ P Z+. If p - t (si) of the partite sets have i (si) vertices and the remaining t (si) partite sets have j (si) vertices we write K [p-t](i) #t (j) for K(i, i,...» i, j, j,..., j). We specifically define K t(i),s(o) = K t(i) = K t(i),o(s) for the positive integers t, i and s, and K o(i),t(j) = K t(j) for the positive integers t, i and j. With this notation, define for each i 5 Z+ by = ^K p(i)'p ^ z + }* I-t is seen that each is an ascending family of graphs. In particular, the family of octahederal graphs is thus denoted $2 T^ e m a in definition of this section is now given.

23 Definition 1.2 Let tt^ tt2» TT be ascending properties of graphs, let i be a positive integer, and assume * ^ ( 1^ # tt2»» tt^) exists. The i-th general ramsey number ^ ( tt^* tt2 «.» n ) is defined to be (tt,, tt-,..., tt, ). 1 2 K This number is sometimes called the ramsey number with respect to Kp(i) ^or t^ie Pro" perties tt^# tt2»...» n. We shall often write r <"l' n2 TT ) for r l (nl' v 2 V * For the integers i and t with t a i we have $i -> $t since K p ^ j <= Kp (t) for every P z+- zt then follows b y anti-monotonicity that if t 2: i then rt (TTi» rr2,, tt^) s tt2»» tt^). The two main consequences of this observation are: (1) If ^ ( t^, tt2»...» tt^) exists then so does ^ ( t^. tt2»...» TT ) for every t i i. (2) The sequence {r t^ni' v 2' * * ' ^^jtsi is a non-increasing sequence of integers (given that it is non-empty) bounded below by 1. Hence there exists xq, the smallest integer i for which r i (TTl' n2' ***' n* = Lim t-*co ^ " l ' n2' **' n^ * led to the following definition. We are Definition 1.3 Let rt^, tt2»..., ir be ascending properties for which r (t^, tt2»...» n ) exists. The ramsey index i(n^» tt2»..., n ) is the least integer

24 17 i for which rj>(tti' n 2' * *' n^ = L *m rt^nl' n2' '**' ^^ * The definition of the ramsey index is slightly restrictive in that it is conceivable that r (tt^» tt2 *» tt^) might not exist while rj_(tti* n2 ' ** ' n^ does exist for some i z 2. In this case a natural alternative definition of a ramsey index could follow. That no such definition is given reflects the fact that the previously described situation never arises in this dissertation. The main wor of Chapter I has been completed. We conclude this chapter with a section of observations and problems. Section 1.4 Closing Comments The original ascending property studied in the context of ramsey numbers is that of graph theoretic inclusion. For the (:sl) graphs G ^ G 2, --- G and the positive integer i, define r i(g i* G 2 ' ***' G^ to be the least integer p such that if Kp (i) = F j then it follows that for some j with 1 i j s we have Gj c P.. That ^ ( G ^ G 2,..., G^) exists for every positive integer i follows directly from antimonotonicity and the fact that r i(g i» G 2 ' **' G^ nown to exist. The existence of the corresponding

25 18 ramsey index denoted i (G^, G 2»...» G^) is thereby guaranteed. We compare and contrast these notions with a "ramsey number" defined by Beinee and Schwen. In [ 2 ], for the positive integers m and n, Beinee and Schwen define (in the current notation) the bipartite ramsey number R(m,n) to be the least positive integer i such that if K 2 ^ = F 1 * F 2 then either K(m,n) c F 1 or K(m,n) c F 2 * It is submitted here that R(m,n) should be thought of as a (bipartite) ramsey index (in the sense just described) rather than a ramsey number for the following two reasons. First, since all other nown ramsey numbers (with a single possible exception to be discussed presently) have been defined b y minimizing the number of partite sets in a factored graph, it is most reasonable that ramsey numbers should continue to enjoy this property. Second, it is an easy consequence of definitions to prove that R(m,n) = i (K(m,n),K(m,n)). Hence R(m,n) fits neatly and naturally into general ramsey theory as a ramsey index. The "single exception" mentioned above is now considered. It seems liely that the most one could expect (within a ramsey context) is to now for the graphs G 1» G 2,..., G^ exactly which complete p-partite graphs K = K ^, n2,..., np ) have the property that if

26 19 K = F. then G. c F.for at least one j (1 * j < ). j=l (This claim is made since one must have nowledge of which edges are present in the factored graph if one is to be able to prove something about the factors. Therefore, in general, the factored graph must be a complete multipartite graph.) Indeed, it may be that most generalized ramsey numbers to date consider only pairs of graphs due to a lac of the above type of nowledge. Faudree and Schelp observed in [17] that in order to find r(p, P, P ) in [18] it is helpful to now n l n 2 n 3 for which integers m and n it is the case that if K(m,n) = F^ F 2 then either Pn c F^^ or PR ^ c F 2. (The graph Pn is the path of length n-1 on n ver- \.:es.) For the ordered graphs ( G ^ G ^ they defined (in [17]) the Ramsey bipartite number pair B t G ^ G j ) to be (if it exists) the lattice point (n,m) with n * m such that the statement "whenever K(r,s) = F^ ffi F 2 it follows that G 1 c Fx or G 2 c F 2 " is true if and only if r 3s n and s a m. It is submitted that B(G1,G2) should be thought of as a generalized ramsey index (as defined below) rather than a ramsey number. Definition 1.4 For the (al) graphs G^, G 2»..., G^ let i(g1# G 2,..., G^) = I and let ri (G1, G2,..., Gj^) = pq. For the integer p pq,

27 20 define the generalized ramsey index p ( I ) (6^, G 2,..., Gfc) to be the lattice point (n^, n 2,... np) with n, a n a... a n a p such that the statement "whenever Kfm^, m 2,... mp ) =. F^ it follows that G^ c F^ for at least one j with 1 s j s 1 is true if and only if nij a n^ for 1 s j s p. We note that p(i)(g1> G 2,... G^) exists for every graphs G 1# G 2,..., G^ and for every integer p a pq. In fact, applying "a" to real vectors in the usual way we have (I. I I) a p(i) (Gr G 2,... G ). p copies Moreover, p'(i)(g^, G 2,..., Gfe), when defined in a similar fashion, can never exist if p' < p Q. It is thought that the generalized ramsey index will closely approximate its associated ramsey. index in the case p = pq. It could very well be the case that a detailed study of ramsey indices is needed to enable significant further advances in generalized ramsey theory (with a 3) to occur. This dissertation presents no such study, but ramsey indices will be mentioned from time to time. We end Chapter I by presenting a few problems.

28 21 Problem 1.1 Characterize those ascending properties tt^ > tt2 #...» TT^ for which (tt^, tt2,...» tt^) exists (i 2; 2) if and only if r (r^, tt2,..., it^) exists. Problem 1.2 Conduct an in-depth study of the ramsey index. Try to apply the results to obtain progress in generalized ramsey theory. Problem 1.3 For the positive integers i, m and n define r^(m,q(n)) to b e the least integer p such that whenever Kp(i) = p2 it: necessarily follows that Km c or else E (F2 ) 2 n. It is a direct result of Turdn's theorem (see [1 / p. 237]) and elementary calculus that r(m,q(n)) = c(m-l) + s where c = {(l + VI + 8n/(m - l))/2} - 1 and s = (n/c - (c - 1) (n - l)/2). Extend T u r a n s theorem in such a way that a corollary of the new result yields the value of r ^ m. q C n ) ) for all i Z+.

29 CHAPTER II ON SUBGRAPH RAMSEY NUMBERS FOR THE REGULAR COMPLETE MULTI-PARTITE GRAPHS For the (^1) graphs G2»...», it follows from anti-monotonicity that G 2' * * " ^ r(g1# G 2,..., G^) for every i Z+. It is natural to see a criterion which produces a lower bound for ri(gi» G 2' * *' G^ for every positive integer i. Section 2.1 maes progress toward finding such a criterion and offers a conjecture whose validity would establish the criterion. Section 2.2 presents special cases which support the conjecture. One of the formulas of Section 2.2 finds all i-th ramsey numbers for every finite collection of stars. Section 2.1 Lower Bounds Recall that for the (il) graphs G^, G 2>..., G^ and the positive integer i, we define ri(g i* G2' to be the least integer p such that whenever K /. v = F., then for some j with 1 s; j it P W j=l 3 follows that Gj c Fj. This number is nown to exist. Hence the corresponding ramsey index, the least integer 22

30 23 I for which r j (<*]_» G 2, ***' G^ = r i*g l' G 2' ***' G ^ ' denoted i(g1# G 2,..., G^), exists. If each graph G.. is the complete graph Kn we define r i<n l' n 2 V = ri (Kn i ' Kn 2' X m A n2 V - itkv K n 2... In Section 2.1 we see a lower bound for G2, ***' G^ in terms of ramsey numbers of simpler yet closely related graphs. The following theorem is a lower bound theorem of the type desired which allows us in general to consider only graphs G^ having no isolated vertices. It is divided into two parts to simplify its statement. Theorem 2.1 (part a) Let i be a positive integer, and let G^, G 2,..., G^ be (il) graphs, some of which are empty. In fact (by symmetry) assume that G 1 = m*^ is an empty graph of smallest order from among the graphs G^, 1 i j s. Then r± (Gr G 2...G ) = (m/i). Proof: If K {m/i) (i) * p j ' then v (F-l) = (m/i}*i ;s m. Therefore ml^ c F1, establishing (m/i} as an upper bound.

31 24 Now consider n = (m/i} - 1 and the factorization K /., = F. where only F. has edges. Since n*i < m n n j j=l ^ then if V (G^ ) s m it follows that Gj F j for every j with 1 s j. (In particular F^. ) If 2 ^ j s and V (G j ) < m, then je (G j ) s 1 implying Gj czf Fj. Hence ^ ( G ^ G 2,...» G ) > (m/i) - 1 Theorem 2.1 (part b) Let i be a positive integer, and let G^, G 2,...» G be (si) graphs, none of which has isolated vertices. For each j with 1 s j s let nij be a non-negative integer, and let Hj = m^i^ U Gj. (If ntj = 0 then Hj = Gj. ) Let p = max-[ V (H j ) : 1 * j s }. Then H 2,..., B^) - max{ri (G1, G 2,..., G]t),(p/i}}. Proof: Let m = max ri (G1# G2,..., G^J^p/ijj-. There are two cases. Case 1: m = r ± ( G ^ G 2,..., G ). By definition there is a factorization K [m-1] (i) =. Fj in which Gj ^ Fj for every 3 * I s j s. Since Gj c Hj then Hj Fj for every j, 1 s j s. Case 2: m = (p/i) It is without loss of generality that we assume m = ( V (G1 ) + m ^ / i }. As in part a, let

32 25 K r i w \ = F.4 Lm-lJ (i; 3 where F. 3 is empty if j i 2. For each j with 2 <s j it follows that 9? [(m-l)i]k^ since JE(H^) i 1. Moreover, H i ^ F i since V(F1 )» [m - 1 ] i < p = V (H1). It therefore follows that H2 #..# H^) ss m. Conversely, for any factorization Km (i) =. ^Fj there is some j with I s j s such that G. c F. o o (by continuity in the integers) since m ri<g i' G 2 ' G^ * since we also have m a tp/i) * (IV(G. ) + m. )/i, then v(f. ) = m - i * v ( G. ) + m.. Jo Jo Jo Jo Jo Thus m. K 1 c F. - V (G. ) from which we have H. c F., 3o ^o Jo Jo Jo concluding the proof of Theorem 2.1 (parts a and b) We may now concentrate exclusively on finding i-th ramsey numbers for graphs without isolated vertices. Theorem 2.1 essentially states that the i-th ramsey number for graphs having isolated vertices is bounded below (due to anti-monotonicity) by the i-th ramsey number of the "cores" of the graphs. A result is now presented which maes a similar type of statement. Note that for the graph G we denote the clique number (the order of the largest complete subgraph of G) by <u(g).

33 26 Theorem 2.2 (lower bound) Let i be a positive integer, let G^, G 2#..., be (s:l) graphs, and let w(gj) = for every j (1 * j * ). Then ^ ( G ^ G 2,..., G]c) a r(n1# n 2,..., i^). Proof; Let ^ ( n ^ n 2,..., n^) = p. It follows that there is a factorization K [p_x] (i) =, F j for Kn gf Fj for every j with 1 s j s. Since K c G. it follows that G. F. where 1 s j a. Hj j j j This implies r x(g x' G 2 ' ' G]c^ ^ P Ifc suffices to show that p s r(n^, n 2,..., n^). To establish that p a r(n1# n 2, n^), let K = F. be any arbitrary factorization of K into P j=i 3 P factors. Label the vertices of Kp (and hence of each factor) with the elements of' {v^, v 2»..., vp ). We construct Kp ^ from Kp by the following process. To obtain the vertex set of Kp(x) replace each vertex Vj of Kp with a copy of, denoted here b y (K7)Vj. The vertices ug 6 (Kx)vt u a ^ ^Ki^vb form an edge of Kp ^ if and only if vt ^ vb * Now for each j with 1 s j s: we construct the graphs Fj from the graphs Fj in a similar manner. That is, the vertex set of Fj is formed b y replacing

34 27 each vertex v fc of Fj with a copy of denoted (K-)v t where 1 t p. The vertices u s 6 (K i^v t and u & are adjacent in if and only if v fc and vfe are adjacent in Fj. It now follows that K = FI. By the defij=i J nition of p, we may assume, without loss of generality, that Kn * F i * No two vertices f this copy of Kn can come from the same partite set of Kp(jj Hence we can form the set U = {u^, u 2,» up } of vertices so that (i) each partite set of Kp ^ ) contributes exactly one vertex to the set U and (ii) V(K ) c U. By re- 1 labeling if necessary, we specify u fc (K ^)vt for each t with 1 t p. Recall that V(Kp ) = (v^, v 2,..., vp ). Define the graph G by (i) V(G) = U (ii) ugut E(G) if and only if ugut E (F ). It is a routine matter to verify that the map ufc -» vfc (for every t with 1 t p) yields an isomorphism for the graphs G and F 1. Hence c G = F.^. Since K = 0 F. is an arbitrary factorization, P j=i J and since there is some j for which Kn c Fj with 1 j, then the conclusion is that

35 28 r(n^, n 2,..., r^) s p which concludes the proof Two immediate corollaries of Theorem 2.2 concern the classical ramsey problem and the ramsey index. Corollary 2.2 A For the positive integers i and n^, n 2,...» n^ it follows that ri(ni» n2 ' = r (n i' n2* ***' "^ * Proof; By anti-montonicity, r^(n^, n 2,...» n^) r(n1# n 2,..., n^). By Theorem 2.2 r i(n ]/ n 2' ***' ^ * r(u)(kn 1 )' cu(kn 2 )' (u(kn )) = r(nl' n 2* n ) * Corollary 2.2 B For the (al) integers n^» n 2,..., n^ it follows that i O ^, n 2,..., n^) = 1. It is natural to as if the lower bound is always attained. That is, given the graphs G^, G2,...» G^» and given I = i C G ^ G 2,..., G^), is it the case that ri (G l» G 2,..., G^) =r(d)(g1 ), tu(g2 ),..., cufg^))? It is easily seen that r(k2,c,.) = 5 and r i (K2,C5 ) = 3 whenever i ^ 2. In general then, the answer is no. We may note however that

36 29 rj {K2.C5 ) = r(x (K2 ),X (C5 )) = r (2,3) = 3. Erdos has conjectured [4,9] that for every graph G, r (G,G) s r(x (G),x (G)), which (by anti-monotonicity) is consistent with the following conjecture. Conjecture For the (*1) graphs G 1# G 2,..., Gj^ it holds that r i (G1, G 2,..., G^) s r f x ^ ), X (G2 )...x{g )} for every i 6 Z+. Moreover, if I = i(g1, G2, G^) then ri (G1, G 2 G^.) - r ( x (G1), X «V ' * x t ^ ) ) - The next section presents some formulas in support of the conjecture. Section 2.2 Exact Results The next result presents the exact value of i-th ramsey numbers for many pairs of graphs. We note that for the graph G, the independence number g (G) is the maximum cardinality among the independent sets of vertices of G. For the disjoint graphs G and H we denote the join of the graphs b y G + H where

37 (i) V(G+H) = V(G) U V(H) and (ii) E (G+H) = E(G) U E (H) U {uv u V (G) and v 6 V(H)). Finally, if U c V(G) then the subgraph induced b y U denoted <U>Q where (i) V(<U>G ) = U and (ii) vertices u and v from U are adjacent in <U>G if and only if uv E ( G ). That H is an induced subgraph of G is denoted by H < G. Note that if H < G and G = F., then j=l 3 H = <V(H)>. j=l F j Theorem 2.3 Let n Z+, let G be a graph, and let 1 = i(k(l,n),g). Then r^(k(l,n),g) = y(q ) Proof; For each graph H define M (H) = 0 (H) (n - 1) (X (H) - 1) and 1(H) = B (H) (M (H) + 1). It suffices to show r^(k(l,n),g) = y(g) for every positive integer i a: I (G). This fact is established by induction on y (G) = p.

38 31 If * = p = 1, then G = for some s Z+. By Theorem 2.1 a, ri (K(l,n),G) = (s/i} whenever i 6 Z+. In particular ri (K(l,n),G) = 1 = X (G ) for every i s s. Since 1(G) = I(Kj) = s, then the induction is anchored. We assume p a 2 and that X (G) = p. Also assume that whenever H is a graph with 1 X (H) s! p - 1 it follows that r.(k(l,a),h) = X (H ) for every positive integer i a: 1(H). It is shown that r i (K(l,n)#G) = x(g ) whenever i a I(G). Let i be an integer for which i i I(G). Since p s 2, consider K [p_i] (j.) = Fi F2 wl*ere F1 is empty. Since (E (F^) < E(K(l,n)) then K(l,n) ^ Since x(p 2 ) < x(g ) then G F 2 - Hence ^(Kll.nJ.G) > p - 1 = x(g ) - 1 To show ri (K(l,n),G) s X (G), let K X (G ) (j.) P i F 2 with K(l,n) $zf F 1.It suffices to show G c F 2. To this end let a,x (<3)-coloring of G be. specified, and let the resulting color classes be v 1# V 2,..., V ^ G j. Let H = G - V 1. Then G c H + 0 (G) so it suffices to show H + S ( Q ) ^ c F2 to conclude G c F2 * Let the partite sets of K.w., be u,, U,,..., U lrs and consider X (G)(i) 1 2 X (G) the factorization K [X CG)-1] (i) * K x (G)(i) - D l = (F1 - V <F2 - X> We have x (H ) = x(q ) - 1 * 1 and H < G so 0(H) s 0 (G). It follows that 1(H) < 1(G) s i.

39 Moreover b y the inductive hypothesis r^(k(l,n),h) = x(h) = x(g ) " 1 - Since r.(k(l,n),h) = x (G) - 1 and since K(l,n) ^ F^ -, then it follows that H c: f 2 - U^. We now show H + B ( G ) c f 2 to conclude the proof. Recall that juj - i * I (G) = B (G) (M(G) + 1). Hence we may partition U.^ into M(G) + 1 sets W1# W 2,..., WM (G)+1 where w.. * 0(G) for every j with 1 s j s M(fl) + 1. If for each j (with 1 j M(G) + 1 ) there exist vertices w.. and u^ 6 V(H) for which w^u.. 6 E(F1 ), then the average degree in F^ of a vertex of H is at least M(G) + 1. M (G) + 1 1V (H) 2 B (H) x (H) * M(G) + 1 _ S (G) (n-1) (*(G)-1)_+ 1 0 (G) (X (G)-1) 0 (G) (X (G) - 1) The above assumption is thus seen to imply K(l,n) c F 1. This contradiction implies that there is some j with 1 j s M(G) + 1 for which every vertex of Wj is adjacent in F 2 with every vertex of H, which in turn implies that H + 0 (G)K^ c F2, concluding the proof The following corollary follows implicitly from the proof of Theorem 2.5.

40 Corollary 2.3 A Let n be a positive integer, and let G be a g r a p h. Then i(k(l,n),g) [0 (G) ]2 (n - 1) (X (G) - 1) + 0 (G). The bound from Corollary 2.3 A on i(k(l,n),g) is sharp in the cases (i) G = Ks for some s Z+ and (ii n = 1 and G = Kp for some p Z+. In general the bound is not particularly good. For example, it will b shown in Chapter III that i(k(l,3),k4 - e) = 4. Howev [0 (K4-e)]2 (3-l) (X (K4-e) - 1) + 0 (K4-e) = = 18 We note that x(k U «n )) = 2 for n 6 Z+ and that r(2,m) = m for m Z+. Hence Theorem 2.3 states that r-j. (K(l,n),G) = r (\(K(l,n)), x (G)) which supports the conjecture of the last section. We now begin the tas of establishing the final main result of Chapter II. For the graph G, the edge-chromatic number is denoted by xi (Q ) 311(1 the maximum degree is denoted by A (G). The following five results will be used in the proof of the main theorem. Theorem A (Vizing [35]) For every graph G it follows that A (G) X l (G) * M Q ) + 1 -

41 Theorem B (Lasar and Hare [24]? Himelwright and Williamson, unpublished) Let p and i be positive integers. Then > i < p (i)} = W if and only if p*i is even. Theorem C (Petersen [30]) Let p and i be positive integers with p z 2. Then K,.x is 2-factorable if and only if (p-l)i is p U ) even. Lemma 2.4 (a) Let n 1# n 2,..., be (asl) positive integers, let S = Y (n^ - 1), and let G be a graph such that j=l XX (G) S. Then there is a factorization G = H. with j=l 3 MHj) s n. - 1 for each j with. 1 s j i. Proof: Since Xi^G) s s there exist graphs F^, F 2 /...» Pg (obtained from the x^^g) edge color classes of some ^ ( G ) - e d g e coloring of G) such that (i) V(Fj) = V(G) for each j with 1 j S, (ii) A(F.) 1 for each j with 1 j S, and S (iii) G = F.. j = i 3

42 35 Partition {P1# F2,...» Fg } into sets U^, U 2 #...» such that Uj = nj 1 for every j with I s j s. For each j (with 1 s j i ) define H. = F.. It 3 V ui3 follows that A (Hj) s IUj I = nj - 1 for every j with S 1 s j s. Moreover, G = F. = H. j=l 3 j-1 3 Lemma 2.4 (b) Let i,, and n1,n2,..., n^ be positive integers. Then I "j-1 * ({(* +I <vi>m ^ 5=1 5=1 with equality if and only if i (n^ -.1). j=l Proof: Let S = ^ (n^ - 1), and let l + S = t * i + m j=l 3 with O s m s i - 1. We proceed by cases on the value of m to show that S a ({(1 + S)/i} - l)i with equality if and only if i S. Case 0: m = 0. We have 1 + S = t*i.it follows that S a 1 + S - i (with equality if and only if i = 1) = t i - i - (t - 1 ) *i = {[ (1 + S)/i} - l)-i.

43 36 Equality holds if and only if i = 1. That is to say, equality holds if and only if ijs and i S + 1. Since i S + 1 then equality holds if and only if i S. Case 1: m = 1. We have 1 + S = t * i + 1 so that S = t*i. Since i S, it must be shown that equality holds. Note that S = t-i = ((1/i + t} - 1 ) -i = (( (1 + t-i)/i} - 1 ) -i = (((1 + S)/i} - l)-i. Case 2: 2 s m s i - 1. We have l + S = t * i + m with 2 m i - 1 which implies S = t*i + m' with l m ' i - 2. It follows that i^s so that strict inequality must be shown. Note that S = t*i + (m - 1) > t*i «((t + m/i} - 1 ) -i = ({(1 + S)/i} - l)-i For the positive integers i,, and n^, nj,...» let t be the number of n^ that are even, let S = J* (n^ - 1), and define j=l D

44 (3) t is positive; (2.1) otherwise. Define r^ (n^, n2,..., n^/*/) to be t*1 least positive integer p such that whenever Kp(i), F j then there is some j with 1 j for which the star graph K(l,nj) c Fj. A complete evaluation of this ramsey number for stars is now presented. Theorem 2.4 Let i,, and n^, n 2,..., n^ be positive integers. Let t be the number of Uj that are even, let S = ^ (n j - 1) * and define 8 as in (2.1). j=l Then ^ ( n ^ n 2,... n^/*/).= ((1 + S)/i) + 8. Proof; Let p = {(1 + S)/i}. It is first shown that r ^ n - ^ n 2,..., n/*/) * P + 1 Let K [p + 1 j (i) =. Fj 2111(3 assume that K(l,nj) gzf Fj for every j with 1 j 3. Then A(Fj) nj - 1 for every j with 1 j. For every j with 1 j, the average value of the degree of a vertex of Fj does not exceed S/. However, this value is nown to be

45 given by p - i A = C (1 + S ) / i } - i A i 1 A + S A n.) with 1 j. Hence, in general, r ^ n. ^ n 2, * P + 1. Now in particular, let (1), (2), and (3) of (2.1) be satisfied. It is shown that n 2 ' ** ' * P«Let K = F. and assume K(l,n.) c F. for each j P U ) j=1 J J J with 1 j. It follows that ielkp(i)>l = I le(pj>i j=l 4 2 I IvtfjJlrMFj) (2.2) j=l * I(Hj - X) j=l = ^^ (p - l)i (by Lemma 2.4 (b) and (2)) (! > 2 = E(Kp(i)) Hence the inequalities of (2.2) are equalities. In particular, each Fj must be regular of degree n^ - 1.

46 39 By (1) p*i is odd and by (3) we may assume n^ is even. Hence the assumption forces the conclusion that is a graph of odd order which is regular of odd degree. This contradiction implies K(l,nj) c: Fj for some j with 1 s j s. We have established that r i(n i» n 2 ' s P+ 0 * To show that n 2,..., n^) > p we proceed by cases on the value of 9. Case 0: 9 = 0. It follows that (1) p*i is odd, (2) i S, and (3) t is positive. From (1) both p and i are odd. If p = 1 the claim surely is true; hence assume p a 3. It suffices to show the existence of a factorization K. l W n =» F. [p-lj (l) j=1 2 where A(F.) n for each j with 1 j s. By an application of Lemma 2.4 (a) it suffices to show ^K [p-i](i)^ * S * However, we have S = (p - l)*i (by Lemma 2.4 (b) and (2)) a (p - 2)-i + 1 = A ( K [p-l] (i)} + 1 * * l (K[p-l] (i)} (by :6111 A ) * Case 1: 0 = 1. It follows that either (1)* p*i is even, or (2)1 i.j's, or (3) t = 0.We show the existence of a

47 40 factorization Kp(iJ = ^Fj with A (Fj) as nj - 1 for every j such 1 s j s. If p*i is even then *i<w MW (by Theorem B) = (p - 1 ) -i ss S (by Lemma 2.4 (b)). An application of Lemma 2.4 (a) demonstrates the existence of the desired factorization. We may assume p*i is odd. If ij's then by Lemma 2.4 (b) S > (p - 1)-1. It follows that S i (p - 1) -i + 1 = M K p ( i > ) + 1 1*1«W Again the desired factorization exists as a consequence of Lemma 2.4 (a), and we assume ijs. In case t = 0 we proceed in a slightly different manner. Since p*i is odd then Kp ^ is regular of the even degree (p - 1) *i. There is nothing to prove if p = 1 so we assume p i 3. From Theorem C we conclude that Kp(jj i-3 2 -factorable into (p - 1) *i/2 2-factors. Since i S then (p - 1 ) *i/2 = S/2, by Lemma 2.4 (b).

48 41 Now t = 0 so every is odd. Hence we can partition this set of 2-factors into the classes U lf U 2,..., so that Uj = (n^ - l)/2 for every j with 1 s j s:. Define the graph Fj to b e the "edge sum" of of the 2-factors in Uj for each j with 1 <,j s. It follows that each Fj is regular of degree 2 Uj = nj 1 and that Kp (i) = Fj We have established that ri(n]/ n2' ** ' ^ * ^ > P which concludes the proof of the theorem The cases i = 1 and i = 2 are stated explicitly. Corollary 2.4 A (Burr and Roberts [6]) Let and n1# n2,..., be positive integers, and let t be the number of nj that are even. Then r (n^, n 2,..., i\/*/) ^ f o if t is even and positive Y n *4 j3 1 ^ Q i otherwise. Proof: Conditions (1), (2), and (3) of (2.1) are true if and only if t is even and positive in the case i = 1 Corollary 2.4 B (the octahedral case) Let and n 2,..., n^ be positive integers.

49 42 Then r 2<nl' n 2 V * / ) - {(l + I O j - l))/2} + 1 j=l Proof: By the definition of 8, it follows that 0 = 1 whenever i is even define For the positive integers and n^, n2,..» i(n1# n 2,..., V'*/) = i(k(l,n]l), K(l,n2 ),..., K d ^ ) ). Corollary 2.4 C Let and n^, n 2, n^ be positive integers, and let I = i(n1# n 2, i\/*/) Then n 2, n^/*/) = 2 * P roof: By the definition of 8, it follows that 0 = 1 if i ^ S + 1 whence ^ ( n ^ n 2,...» i\/*/) ^ 2 of course ri(n i' n 2 ' ***' > ^ for every i 6 Z+ Corollary 2.4 D Let and n ^ n2,..., n^ be positive integers, Then i O ^, n 2,..., n^/*/) = 7 (n.. - 1) + 1. P r o o f : Let S = (n^ - 1). If S = 0 the result is j=l j=l evident, so tae S to be positive. By Theorem 2.4,

50 43 rs (nlf n 2,..., n^/*/) = { (S + 1)/S} + 1 = 3 since condition (1) of (2.1) is false. By anti-monotonicity and Corollary 2.4 C it follows that i(nlf n 2,..., l\/*/) > S. However, rs+l^n i' n 2' * *' n//*'/) = ( (S + 1)/(S + 1)) + 1 = 2 since condition (2) of (2.1) fails to be true. Then b y definition and Corollary 2.4 C we have i(n^, n 2,..., nfc/*/) = S + 1 It seems reasonable to assert that examples of particular ramsey numbers with respect to the graphs Kp(jj are needed before much more progress can be made. In order to provide evidence for conjectures and possible base cases for inductive arguments, over 90% of the "small" i-th ramsey numbers and over 90% of the "small" ramsey indices are computed in Chapter III. Chapter II is concluded with some problems. Problem 2.1 Support or disprove the conjecture of Section 2.1 by finding r^(c^,cg) for every i Z+. Problem 2.2 Extend Theorem 2.3 so that the star is replaced by an arbitrary tree. Extend this new result so the tree

51 44 is replaced by a forest. Mae a final extension which solves E r d S s 1 conjecture given in Section 2.1. Problem 2.3 The next most accessible formulas would seem to come from paths [19], matchings [15], and stars and matchings [16]. Find the ramsey numbers of these families of graphs with respect to the symmetric complete p-partite graphs. (The graph nk2 is a matching.)

52 CHAPTER III MOST SMALL i-th RAMSEY NUMBERS In order to establish information for future study, most of the "small" i-th ramsey numbers are determined in Chapter III. Section 3.1 Preliminary Results The motivation for the wor of this chapter is given b y Harary [21, p. 11]. "It is useful to obtain the ramsey numbers for small graphs in order to have data for maing conjectures. Also, as Burr [5] observed, these small cases often provide the starting point for inductive proofs, but must be proved independently." As an extension of a definition b y Chvdtal and Harary [12], the small i-th ramsey numbers are those ramsey numbers r^(f,g) for which i Z+ and F and G are graphs of order less than five having no isolated vertices. The small ramsey indices are those ramsey indices i (F,G) for which ri (F,G) is a small i-th ramsey number. If F = G, then ri (G,G) is called a diagonal i-th ramsey number. 45

53 46 The need for the data supplied by small (1-st) ramsey numbers has generated many papers. For example Harary and Chvatal found all small (1-st) diagonal ramsey numbers in [1 2 ] and all small (1-st) non-diagonal ramsey numbers in [13], Also, Clancy [14] has recently determined most of the (1-st) ramsey numbers r(f,g) where F and G are graphs of orders at most four and five (respectively) having no isolated vertices. Chapter III continues the trend outlined in the preceding paragraph b y finding nearly all small i-th ramsey numbers (as well as their corresponding ramsey indices). The only unsolved cases are ri (C4,K4 ), ri (K4-e,K4~e ), and r^(k4 ~e,k4 ) ; i a 2. The graphs to be considered and their symbolic names are listed in Figure 3.1. A summary of the ramsey numbers found in this chapter is given in Figure 3.2. With three exceptions (which are each indicated by "?") the ramsey index i(f,g) is the largest value of i for which r^(f,g) is given in Figure 3.2. The small ramsey indices are explicitly given in Figure 3.3. It may be noted that for the graphs G to be considered in this chapter tu(g) = x(g) Hence it may not be surprising that for I = i (F,G), it follows that r ].(F,G) = r(x (F),x (G)) every nown case for I in Chapter III.