Assorted results in boolean function complexity, uniform sampling and clique partitions of graphs

Abstract

This thesis consists of three disparate parts. In the first, we generalize and extend recent ideas of Chiarelli, Hatami and Saks to obtain new bounds on the number of relevant variables for a boolean function in terms of its degree, its sensitivity, and its certificate and decision tree complexities, and we also sharpen the best-known polynomial relationships between some of these complexity measures by a constant factor. In the second part, we show that the Partial Rejection Sampling method of Guo, Jerrum and Liu can solve a handful of natural sampling problems that fall outside the guarantees of the authors' original analysis. Finally, we revise and make partial progress on a conjecture of De Caen, Erdős, Pullman and Wormald on clique partitions of a graph and its complement, building on ideas of Keevash and Sudakov.