We study open sets P in normed spaces X attaining a large volume while avoiding pairs of points at integral distance. The proposed task is to find sharp inequalities for the maximum possible d-dimensional volume. This problem can be viewed as an opposite to known problems on point sets with pairwise integral or rational distances.
In answer to a question of Macpherson and Neumann related to the classiication of maximal subgroups of innnite symmetric groups, we characterize set ideals on an innnite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Speciically, for an ideal I containing all subsets of cardinality less than jXj, the stabilizer S fIg is maximal… (More)