Combinatorial principles from adding Cohen reals

Abstract. We first formulate several “combinatorial principles” concerning κ × ω matrices of subsets of ω and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any ground model V, provided that the parameter κ is an ω-inaccessible regular cardinal in V. Then in section 4 we present a large number of applications of these principles, mainly to topology. Some of these consequences had been established earlier in generic extensions obtained by adding ω2 Cohen reals to ground models satisfying CH, mostly for the case κ = ω2. 1

Some Applications of Laplace Eigenvalues of Graphs

In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs