The Ramsey number r(H) of a graph H is the minimum positive integer N such that every twocoloring of the edges of the complete graph KN on N vertices contains a monochromatic copy of H . A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erdős in 1975 conjectured that for each positive integer d there is a constant cd such that r(H) ≤ cdn for every d-degener...

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...

‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

The bipartite Ramsey number b(m,n) is the smallest positive integer r such that every (red, green) coloring of the edges of Kr,r contains either a red Km,m or a green Kn,n. We obtain asymptotic bounds for b(m, n) for m ≥ 2 fixed and n →∞.

For a positive integer n and graph B, fa (n) is the least integer m such that any graph of order n and minimal degree m has a copy of B. It will be show that if B is a bipartite graph with parts of order k and i (k-1), then there exists a positive constant c, such that for any tree T of order n and for any j (0-j _-(k-1)), the Ramsey number r(T, B) < n + c-(fe (n)) 't(k-1) if A(T)-(n/(k-j-1))-(...

We generalize the concept of pattern occurrence in permutations, words or matrices to that in n-dimensional objects, which are basically sets of (n + 1)-tuples. In the case n = 3, we give a possible interpretation of such patterns in terms of bipartite graphs. For zero-box patterns we study vanishing borders related to bipartite Ramsey problems in the case of two dimensions. Also, we study the ...

In this paper a concept Q-Ramsey Class of graphs is introduced, where Q is a class of bipartite graphs. It is a generalization of wellknown concept of Ramsey Class of graphs. Some Q-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that T 2, the class of all outerplanar graphs, is not D1-Ramsey Class (Theorem 3). This results leads us to the concept of acyclic reducible bounds...

Let G1 and G2 be two given graphs. The Ramsey number R(G1,G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or G contains a G2. A complete bipartite graph K1,n is called a star. The kipas ̂ Kn is the graph obtained from a path of order n by adding a new vertex and joining it to all the vertices of the path. Alternatively, a kipas is a wheel with one edg...

We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small bandwidth. Our main theorem implies that there exists a constant c such that for every ∆, there exists β such that if G is an n-vertex graph with maximum degree at most ∆ having a homomorphism f into a...