Let G and H be graphs. A graph with colored edges is said to be monochromatic if all its edges have the same color and rainbow if no two of its edges have the same color. Given two bipartite graphs G1 and G2, the bipartite rainbow ramsey number BRR(G1; G2) is the smallest integer N such that any coloring of the edges of KN;N with any number of colors contains a monochromatic copy of G1 or a rai...

Let $H \xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H G \}$, and the bipartite $br_{\Delta}(G;s)$ \textrm{ } \chi(H) = 2 \}$. In this note, we show $r_{\Delta}(K_{m,n};s)$ linear on $n$ with fixed $m$. We also evaluate for paths other trees.

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.

A partial ordering P is chain-Ramsey if, for every natural number n and every coloring of the n-element chains from P in finitely many colors, there is a monochromatic subordering Q isomorphic to P. Chain-Ramsey partial orderings stratify naturally into levels. We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a c...

In this paper we survey author’s recent results on quantitative extensions of Ramsey theory. In particular, we discuss our recent results on Folkman numbers, induced bipartite Ramsey graphs, and explicit constructions of Ramsey graphs.

The Zarankiewicz number z(m,n; s, t) is the maximum number of edges in a subgraph of Km,n that does not contain Ks,t as a subgraph. The bipartite Ramsey number b(n1, · · · , nk) is the least positive integer b such that any coloring of the edges of Kb,b with k colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1 ≤ i ≤ k. If ni = m for all i, then we denote this ...

Let G be a bipartite graph, with k j e(G). The zero-sum bipartite Ramsey number B(G; Z k) is the smallest integer t such that in every Z k-coloring of the edges of K t;t , there is a zero-sum mod k copy of G in K t;t. In this paper we give the rst proof which determines B(G; Z 2) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G; Z 2) can be deduce...