A graph is properly edge-colored if no two adjacent edges have the same color. The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number ARs(H) of H. This number in offline and online setting is investigated here.

Ramsey theory deals with the problem of finding structured configurations in suitably large but possibly disordered sets. The nature of the desired configurations can range from complete subgraphs of a graph to arithmetic progressions in Z to solutions of equations, such as x+y = z, in a countable commutative semigroup. In this paper we deal with configurations consisting of finitely many colli...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai-Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of ...

The Ramsey number R(C4,Km) is the smallest n such that any graph on n vertices contains a cycle of length four or an independent set of order m. With the help of computer algorithms we obtain the exact values of the Ramsey numbers R(C4,K9) = 30 and R(C4,K10) = 36. New bounds for the next two open cases are also presented.

For graphs G and H, the size balanced multipartite Ramsey number ) , ( H G m j is defined as the smallest positive integer s such that any arbitrary two red/blue coloring of the graph s j K × forces the appearance of a red G or a blue H . In this paper we find the exact values of the multipartite Ramsey numbers ) , ( 2 3 nK P m j and ) , ( 2 4 nK P m j .

We prove induced Ramsey theorems in which the induced monochromatic subgraph satisfies that some of its partial automorphisms extend to automorphisms of the colored graph.

For any graph F , V (F ) and E (F ) will denote the vertex and edge sets of the p graph F , also let n (F ) = | V (F ) | and e (F ) = | E (F ) | . The graph F  denotes the com lement of F . A graph F will be called a (G , H )−good graph, if F does not contain a ( G and F  does not contain H . Any (G , H )-good graph on n vertices will be called G , H , n )−good graph. The Ramsey number R (G ,...

We prove that s r ( K k ) = O 5 / 2 $s_r(K_k) O(k^5 r^{5/2})$ , where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erdős and Lovász in 1976, which defined as smallest minimum degree of a graph G $G$ such any $r$ -colouring edges contains monochromatic $K_k$ whereas no proper subgraph has this property. The construction used our proof relies on group theoretic model generalised quadran...