All triangle-graph ramsey numbers for connected graphs of order six

The Ramsey numbers r(K., G) are determined for all connected graphs G of order six. Ramsey numbers for complete collections of small graphs have been studied by several mathematicians. Chv6tal and Harary determined all Ramsey nurnbers for pairs of graphs each with at most four vertices [1, 2]. Clancy has produced a table of Ramsey numbers r(F, G), where F is of order at most four and G is of order five [3]. This table is complete except for five entries. Burr has determined all diagonal Ramsey numbers for graphs of size at most six. Burr's results are quoted in the article on Ramsey theory which appeared in the "Mathematical Games" section of Scientific Afnerican, November, 1977. Ramsey numbers for small graphs have evident usefulness. They provide information which can be used in proving additional results in generalized Ramsey theory, they provide raw material for new conjectures, and they give some insight concerning the complexity of graphical Ramsey theory in general. As matters stand, completion of the table of Ramsey numbers for all graphs of order at most five seems somewhat out of reach, since this would involve, among other things, computing the classical Ramsey number r(Ku)' Doing the sarfle for ali graphs of order at most six seems totally hopeless in the Journal of Graph Theory, Vol.4 {1980) 293-300 o 1980 by John witey & Sons, lnc. 0364-9O24/80/0004-0293s01.00 294 JOURNAL OF GRAPH THEORY absence of some extraordinary new tool. In this note, we determine r(Kr, G) for all connected graphs G of order six. On the face of it, even this project appears to be one of some complexity. one reason is that there are 112 such graphs. However, these 112 graphs can be grouped into six classes corresponding to six possible values of r(K., G) and arguments covering the five classes of noncomplete graphs are surprisingly simple. All proofs will be given in detail. Let V:{tsr,ur,. ,uo} denote a set of vertices. Then [V], denotes the set of all unordered pairs of these vertices. By a two-coloring we mean a partition lvf': (R, B). Equivalently, we ascribe to each edge of the complete graph of order p a color, either red or blue. This two-coloring defines two edge-induced graphs of order p and we use (R) and (B) as symbols for these graphs. Let F and G be graphs without isolated vertices. The statement Kr-+(E, G) means that if lVl:p, then for every possible two-coloring (R, B) of [V]2, either (R) contains (a subgraph isomorphic to) F or (B) contains G. The Ramsey number r(E, G) is the smallest natural number p such that Kp -(E G). properties of (R) and (B) will be denoted in an obvious way. Thus, for example, if u e V, then N^(o) and Nr(o) denote the neighborhoods of u in (R) and (B), respectively. Let X be a subset of V. Then we shall use Xo(u) and X. (o) to denote Np(u)OX and N"(u)nX, respectively. One additional specialized notation will be employed. Let X and Y be nonempty disjoint subsets of V and suppose that lxl is odd. The set of all vertices of y which are adjacent in (R) to more than half of the vertices of X will be denoted Y(X R). Similarly, Y(X B) denotes the set of vertices u e y such that lx"(o)l>+ lxl. Lemma. Let (R, B) be a two-coloring of fV]'z such that (R) contains no Kr. Let X and Y be nonempty disjoint subsets of V such that [Xl is odd. The set Y(X R), consisting of those vertices of Y which are adjacent to more than half of the vertices of X in (R), spans a complete graph in (B). Proot. If lY(X R)l:1,there is nothing to prove. Otherwise, suppose that u and u are distinct vertices in Y(XR) and note that Xo(u)fl X"(a)* Z. Thus uueB since (R) contains no K3. I The simplest example of this lemma is that for any vertex o, N*(u) spans a complete graph in (B). The following example is a bit more interesting and will be used several times in what follows. Corollary. Let (R,B) be a two-coloring of [V]2, where lvl>11. Suppose that (R) contains no Kr, but that (B) does contain a K. Then (B) contains KuPr. Proof. Let X span K, in (B) and let Y : V X. If lY(X R)l > 6, then TRIANGLE-GRAPH RAMSEY NUMBERS 295 (B) contains a K6.Otherwise, Y(X,B)+A,i.e., there is a vertex of Y which is adjacent in (B) to at least three vertices of X. we thus have the stated conclusion. I We are now prepared to prove the main result. Theorem. With the graphs Go through Gp as identified (by their complements) in Figure 1, the following table gives r(Kr, G) for every connected graph G of order six.