Linear-algebraic extremizers for symmetric cancellative pairs

Determine whether, for every positive integer n, the maximum value G(n) of |A| over families A 2^[n] such that (A,A) is cancellative is attained by a family |F_{M,w}| constructed from some k n matrix M over F_2 and vector w F_2^k.

Background

For a symmetric cancellative pair, the paper defines G(n) as the maximum size of a family A of subsets of [n] for which (A,A) is cancellative. The main theorem gives the upper bound G(n) 1.5^n, while Tolhuizen's construction produces candidate extremal families of the form F_{M,w}, consisting of k-subsets whose associated k by k submatrix of a binary matrix M is invertible and whose column sum equals w.

The authors note that computer searches support the linear-algebraic construction for n 8, but do not establish that such a construction is optimal for every n. The unresolved question asks whether every exact maximizer G(n) can be represented in this form for a suitable binary matrix and vector.