 The KS Theorem claims the existence of a set M with a certain property  (i.e. being such that KS1 and KS2 are  contradictory)[6]  and the proof proceeds by explicitly presenting such a set, for different choices of x and y.  In the original KS proof x=3 and y=117. More recently proofs involving less observables have been given by (among many others) Peres (1991, 1995) for x=3 and y=33, by Kernaghan (1994) for x=4 and y=20 and by Cabello et al. (1996) for x=4 and y=18. The KS proof is notoriously complex, and we will only sketch it in Section 3.4. The Peres proof establishes the KS result in full strength, with great simplicity, and, moreover, in an intuitively accessible way, since it operates in three dimensions; we refer the reader to Peres (1995: 197–99). The proofs by Kernaghan and Cabello et al. each establish a contradiction in four dimensions. These are weaker results, of course, than the KS theorem (since every contradiction in 3 dimensions is also a contradiction in higher dimensions, but not conversely). However, these other proofs are very simple and instructive. Moreover, it can be shown (Pavičić et al. 2005) that y=18 is the lowest number for which the KS theorem holds true, so we start by presenting the proof of Cabello and his co-workers in Section 3.2. Finally, in Section 3.5, we explain an argument by Clifton (1993) where x=3 and y=8 and an additional statistical assumption yields an easy and instructive KS argument.