On globally sparse Ramsey graphs

On globally sparse Ramsey graphs

We say that a graph G has the Ramsey property w.r.t. some graph F and some integer r ≥ 2, or G is (F, r)-Ramsey for short, if any r-coloring of the edges of G contains a monochromatic copy of F . Rödl and Ruciński asked how globally sparse (F, r)-Ramsey graphs G can possibly be, where the density of G is measured by the subgraph H ⊆ G with the highest average degree. So far, this so-called Rams...

On Ramsey Numbers of Sparse Graphs

The Ramsey number, r(G), of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph KN on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(...

On the Ramsey Number of Sparse 3-Graphs

We consider a hypergraph generalization of a conjecture of Burr and Erdős concerning the Ramsey number of graphs with bounded degree. It was shown by Chvátal, Rödl, Trotter, and Szemerédi [The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), no. 3, 239–243] that the Ramsey number R(G) of a graph G of bounded maximum degree is linear in |V (G)|. We derive...

On Ramsey Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest F other than a matching, and every graph G containing a cycle, there exists an infinite number of graphs J such that J → (F,G) but if we delete from J any edge e the graph J − e obtained in this way does not have this property. Introduction. All graphs in this note are undirected graphs, without loops and multip...

On Ramsey Minimal Graphs