Aspects of quasi-randomness: VII Maximum cuts and quasi-random graphs.

This is the seventh in a series of memoranda dealing with certain aspects of randomness occurring in the mathematical sciences. The use of "random" structures and "random" algorithms plays an essential role in a variety of disciplines, which includes mathematics, theoretical computer science and communication theory, to name a few. Indeed, Shannon's fundamental results on the existence of graphs on n vertices which have maximum cliques and independent sets of size 2 log n are some of the earliest examples of the power of employing random structures. One of the most fruitful applications of these ideas is the so-called "probabilistic method". Typically what is done in this approach is to prove the existence of some desired object by the following scheme. First, an appropriate (probability) measure is defined on the class of objects under investigation; second, the subclass of desired objects are shown to have positive measure.