Recent developments in graph Ramsey theory

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress

Extremal colorings and extremal satisfiability

Combinatorial problems are often easy to state and hard to solve. A whole bunch of graph coloring problems falls into this class as well as the satisfiability problem. The classical coloring problems consider colorings of objects such that two objects which are in a relation receive different colors, e.g., proper vertex-colorings, proper edge-colorings, or proper face-colorings of plane graphs. A generalization is to color the objects such that some predefined patterns are not monochromatic. Ramsey theory deals with questions under what conditions such colorings can occur. A more restrictive version of colorings forces some substructures to be polychromatic, i.e., to receive all colors used in the coloring at least once. Also a true-false-assignment to the boolean variables of a formula can be seen as a 2-coloring of the literals where there are restrictions that complementary literals receive different colors. Mostly, the hardness of such problems is been made explicit by proving that they are NP-hard. This indicates that there might be no simple characterization of all solvable instances. Extremal questions then become quite handy, because they do not aim at a complete characteriziation, but rather focus on one parameter and ask for its minimum or maximum value. The goal of this thesis is to demonstrate this general way on different problems in the area of graph colorings and satisfiability of boolean formulas. First, we consider graphs where all edge-2-colorings contain a monochromatic copy of some fixed graph H. Such graphs are called H-Ramsey graphs and we concentrate on their minimum degree. Its minimization is the question we are going to answer for H being a biregular bipartite graph, a forest, or a bipartite graph where the size of both partite sets are equal. Second, vertex-colorings of plane multigraphs are studied such that each face is polychromatic. A natural parameter to upper bound the number of colors which can be used in such a coloring is the size g of the smallest face. We show that every graph can be polychromatically colored with \floor{3g-5}{4} colors and there are examples for which this bound is almost tight. Third, we consider a variant of the satisfiability problem where only some (not necessarily all) assignments are allowed. A natural way to choose such a set of allowed assignments is to use a context-free language. If in addition the number of all allowed assignments of length n is lower bounded by $\Omega(\alpha^n)$ (an) for some $\alpha > 1$, then this restricted satisfiability problem will be shown to be NP-hard. Otherwise, there are only polynomially many allowed assignments and the restricted satisfiability problem is proven to be polynomially solvable