On globally sparse Ramsey graphs

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available

On n-stars in colorings and orientations of graphs

An n-star S in a graph G is the union of geodesic intervals I1,…,Ik with common end O such that the subgraphs I1∖{O},…,Ik∖{O} are pairwise disjoint and l(I1)+…+l(Ik)=n. If the edges of G are oriented, S is directed if each ray Ii is directed. For natural number n,r, we construct a graph G of diam(G)=n such that, for any r-coloring and orientation of E(G), there exists a directed n-star with monochrome rays of pairwise distinct colors

The size ramsey number of graphs with bounded treewidth

A graph G is Ramsey for a graph H if every 2-coloring of the edges of G contains a monochromatic copy of H. We consider the following question: If H has bounded treewidth, is there a sparse graph G that is Ramsey for H? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of H are bounded, then there is a graph G with O(| V (H)| ) edges that is Ramsey for H. This was previously only known for the smaller class of graphs H with bounded bandwidth. On the other hand, we prove that in general the treewidth of a graph G that is Ramsey for H cannot be bounded in terms of the treewidth of H alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and H is a tree