We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every twocoloring of the edges of the complete graph KN contains a monochromatic copy of H. A famous resul...

In this paper, we study the generalized Ramsey number r(G, , . . ., Gk) where the graphs GI , . . ., Gk consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G 2 , . . ., G,, are fixed and G, ~_C, or P,, with n sufficiently large . If among G2 , . . ., G k there are both complete graphs and odd cycles, the main theo...

The lower bound for the classical Ramsey number R(4, 8) is improved from 56 to 58. The author has found a new edge coloring of K57 that has no complete graphs of order 4 in the first color, and no complete graphs of order 8 in the second color. The coloring was found using a SAT solver which is based on MiniSat and customized for solving Ramsey problems. Recently Exoo improved the lower bound f...

Following ideas of Richer (2000) we introduce the notion of unordered regressive Ramsey numbers or unordered Kanamori-McAloon numbers. We show that these are of Ackermannian growth rate. For a given number-theoretic function f we consider unordered f -regressive Ramsey numbers and classify exactly the threshold for f which gives rise to the Ackermannian growth rate of the induced unordered f -r...

we canvas a number of past uses of Ramsey sentences which have yielded disappointing results, and then consider three very interesting recent attempts to deploy them for a Ramseyan dialetheist theory of truth, a modal account of laws and theories, and a criterion for the existence of factual properties. we think that once attention is given to the specific kinds of theories that Ramsey had in m...

In this paper we will give all Ramsey numbers r(K 3 ; G) for connected graphs G of order 7 and 8. For smaller orders and not connected graphs of order up to 8, the number of graphs with given Ramsey number is listed for all possible values.

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if any coloring of the edges $F$ in red blue, there is either a induced copy $H$ or blue $G$. For $G$ let $\mathrm{IR}(H,G)$ be smallest number vertices graph such that \longrightarrow}(H,G)$.
 In this note we consider case when star on $n$ edges, large fixed graph. prove $$ (\chi(H)-1) n \leq \math...