No abstract

The statement, that in a tiling by translates of an n-dimensional cube there are two cubes having common (n -I)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a counterexample for this conjecture if and only if the following graphs r,, has a 2" size clique. The 4" vertices of I',, are n-tuples of integers 0, 1, 2 and 3. A pair of these n-tuples are adjacent if there is a position at which the difference of the corresponding components is 2 modulo 4 and if there is a further position at which the corresponding components are different. We will give the size of the maximal cliques of r, for n 2 5.

No abstract

If a finite abelian group is expressed as the product of subsets each of which has a prime number of elements and contains the identity element, then at least one of the factors is a subgroup. This theorem was proved by L. Redei in 1965. In this paper we will give a shorter proof.