We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.
The graph Ramsey number is the smallest integer with the property
that any complete graph of at least vertices whose edges are colored with two colors
(say, red and blue) contains either a subgraph isomorphic to all of whose edges are
red or a subgraph isomorphic to all of whose edges are blue. In this paper, we
consider the Ramsey numbers for theta graphs. We determine , for
. More specifically, we establish that for .
Furthermore, we determine for . In fact, we establish that if is even, if is odd.
For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.
Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.
Answering a question raised by Dudek and Prałat, we show that if pn → ∞, w.h.p., whenever G = G(n, p) is 2-edge-coloured there is a monochromatic path of length (2/3 + o(1))n. This result is optimal in the sense that 2/3 cannot be replaced by a larger constant.As part of the proof we obtain the following result. Given a graph G on n vertices with at least $(1-\varepsilon)\binom{n}{2}$ edges, whenever G is 2-edge-coloured, there is a monochromatic path of length at least $(2/3 - 110\sqrt{\varepsilon})n$. This is an extension of the classical result by Gerencsér and Gyárfás which says that whenever Kn is 2-coloured there is a monochromatic path of length at least 2n/3.
A Note on On-Line Ramsey Numbers for Some Paths
