This is the imaginative title of my PhD thesis. It contains four unrelated pieces of work. (I was warned off using this phrasing in the thesis itself, where the chapters are instead described as “self-contained”.)

The first and most substantial concerns partition regularity. It is a coherent presentation of all of the material from Partition regularity in the rationals, Partition regularity with congruence conditions and Partition regularity of a system of De and Hindman.

The remaining three chapters are expanded versions of Maximum hitting for sufficiently large, Random walks on quasirandom graphs and A note on balanced independent sets in the cube.^{1}

Erratum: the form of Talagrand’s inequality quoted as Theorem 6 in Chapter 4 has an incorrect proof in the book cited. The proof can be modified to prove a slightly weaker result, with 60 replaced by 90 and an assumption that the median of is at least (a constant). This is remarked in the published version of this chapter, Random walks on quasirandom graphs.

^{1} “A note … ” was fairly described by one of my examiners as a “potboiler”. It was also my first submitted paper, and completed in my second of three years. Perhaps this will reassure anybody in the first year of a PhD who is worrying that they have yet to publish.