Induced Ramsey Numbers

We investigate the induced Ramsey number r ind (G; H) of pairs of graphs (G; H). This number is defined to be the smallest possible order of a graph with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with k = jV (G)j t = jV (H)j, we have r ind (G; H) t Ck log q ; where q = (H) is the chromatic number of H and C is some universal constant. Furthermore, we also investigate r ind (G; H) imposing some conditions on G. For instance, we prove a bound that is polynomial in both k and t in the case in which G is a tree. Our methods of proof employ certain random graphs based on projective planes

Probabilistically Checkable Proofs and their Consequences for Approximation Algorithms

The aim of this paper is to present a self-contained proof of the spectacular recent achievement that NP = PCP(log n; 1). We include, as consequences, results concerning non-approximability of the clique number, as well as of the chromatic number of graphs, and of MAX-SNP hard problems

Graphs with large obstacle numbers

Motivated by questions in computer vision and sensor networks, Alpert et al. [3] introduced the following definitions. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if an only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It was shown in [3] that there exist graphs of n vertices with obstacle number at least Ω ( √ logn). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω(n/log 2 n), and (2) the total number of graphs on n vertices with bounded obstacle number is at most 2 o(n2). Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides