Symmetric Difference-Free and Symmetric Difference-Closed Collections of Sets

Abstract

A collection of sets is symmetric-difference-free, respectively symmetric difference-closed, if the symmetric difference of any two sets in the collection lies outside, respectively inside, the collection. Recently Buck and Godbole (Size-maximal symmetric difference-free families of subsets of [n], Graphs Combin. (to appear), 2013) investigated such collections and showed, in particular, that the the largest symmetric difference-free collection of subsets of an n-set has cardinality 2^n-1. We use group theory to obtain shorter proofs of their results.