Abstract

In this thesis we study the independent sets of Knr , the weak product of n complete graphs on r vertices, which are close to be of maximum size. We review the previously known results. For constant r and arbitrary n, it was known that every such independent set is close to some independent set of maximum size. We prove that this statement holds for arbitrary r and n. The proof involves some techniques from Fourier analysis of Boolean functions on Znr . In fact we show that when most of the 2-norm weight of the Fourier expansion of a Boolean function on Znr is concentrated on the first two levels, then the function can be approximated by a Boolean function that depends only on one coordinate. A stronger analogue of this has been proven by Jean Bourgain for Zn2 . We present an expanded version of his proof in this thesis.