Rainbow Ramsey Problems for the Boolean Lattice

We address the following rainbow Ramsey problem: For posets P, Q what is the smallest number n such that any coloring of the elements of the Boolean lattice Bn either admits a monochromatic copy of P or a rainbow copy of Q. We consider both weak and strong (non-induced and induced) versions of this problem.

Ramsey properties of randomly perturbed graphs: cliques and cycles

Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

Optimal Monetary and Fiscal Policy

We construct the Ramsey problem for our model. We use this to examine optimal monetary and fiscal policy.

Ramsey-type problems in orientations of graphs ⇤

The Ramsey number R(H) of a graph H is the minimum number n such that there exists a graph G on n vertices with the property that every two-coloring of its edges contains a monochromatic copy of H. In this work we study a variant of this notion, called the oriented Ramsey problem, for an acyclic oriented graph H~ , in which we require that every orientation G~ of the graph G contains a copy of H~ . We also study the threshold function for this problem in random graphs. Finally, we consider the isometric case, in which we require the copy to be isometric, by which we mean that, for every two vertices x, y 2 V (H~ ) and their respective copies x0, y0 in G~ , the distance between x and y is equal to the distance between x0 and y0.