Ramsey’s Theorem tells us that there are exactly two minimal hereditary classes containing graphs with arbitrarily many vertices: the class of complete graphs and the class of edgeless graphs. In other words, Ramsey’s Theorem characterizes the graph vertex number in terms of minimal hereditary classes where this parameter is unbounded. In the present paper, we show that a similar Ramsey-type ch...

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges produces either red $G$ or blue $H$. We generalize this infinite $H$; particular, we want determine if there minimal $F$. This has strong connections self-embeddable graphs: which properly contain copy themselves. prove some compactness results r...

A local k-coloring of a graph is a coloring of its edges in such a way that each vertex is incident to edges of at most k different colors. We investigate the similarities and differences between usual and local k-colorings, and the results presented in the paper give a general insight to the nature of local colorings. We are mainly concerned with local variants of Ramsey-type problems, in part...

A pair of graphs (Hb,Hr) is highly Ramsey-infinite if there is some constant c such that for large enough n there are at least 2cn 2 non-isomorphic graphs on n or fewer vertices that are minimal with respect to the property that when their edges are coloured blue or red, there is necessarily a blue copy of Hb or a red copy of Hr. We show that a pair of 3-connected graphs is highly Ramsey-infini...

In 1995 Kim famously proved the Ramsey bound R(3, t) ≥ ct/ log t by constructing an n-vertex graph that is triangle-free and has independence number at most C √ n log n. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph Kn into a packing of such nearly optimal Ramsey R(3, t) graphs. More precisely, for any ...

Given a graph H, the size Ramsey number re(H, q) is the minimal number m for which there is a graph G with m edges such that every q-coloring of G contains a monochromatic copy of H. We study the size Ramsey number of the directed path of length n in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every q ≥...

We investigate the minimization problem of the minimum degree of minimal Ramsey graphs, initiated by Burr, Erdős, and Lovász. We determine the corresponding graph parameter for numerous bipartite graphs, including bi-regular bipartite graphs and forests. We also make initial progress for graphs of larger chromatic number. Numerous interesting problems remain open.