Ramsey numbers for sets of small graphs

AbstractThe Ramsey number r=r(G1-G2-⋯-Gm,H1-H2-⋯-Hn) denotes the smallest r such that every 2-coloring of the edges of the complete graph Kr contains a subgraph Gi with all edges of one color, of a subgraph Hi with all edges of a second color. These Ramsey numbers are determined for all sets of graphs with at most four vertices, and in the diagonal case (m=n,Gi=Hi) for all pairs of graphs, one with at most four and the other with five vertices, so as for all sets of graphs with five vertices

R(W5 , K5) = 27

The two-color Ramsey number R(G , H) is defined to be the smallest integer n such that any graph F on n vertices contains either a subgraph isomorphic to G or the complement of F contains a subgraph isomorphic to H. Ramsey numbers serve to quantify many of the existing theorems of Ramsey theory, which looks at large combinatorial objects for certain given smaller combinatorial objects that must be present. In 1989 George R. T. Hendry presented a table of two-color Ramsey numbers R(G , H) for all pairs of graphs G and H having at most five vertices. This table left seven unsolved cases, of which three have since been solved. This thesis eliminates one of the remaining four cases, R(W5 , K5), where a K5 is the complete graph on five vertices and a W5 is a wheel of order 5, which can be pictured as a wheel having four spokes or as a cycle of length 4 having all four vertices adjacent to a central vertex. In this thesis we show R(W5, K5) to be equal to 27, utilizing a combinatorial approach with significant computations. Specifically we use a technique developed by McKay and Radziszowski to effectively glue together smaller graphs in an effort to prove exhaustively that no graph having 27 vertices exists that does not contain an independent set on five vertices or a subgraph isomorphic to W5. The previous best bounds for this case were 27 \u3c= R( W_5 , K_5 ) \u3c= 29

On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page