  In 1986 Lovasz, Spencer, and Vesztergombi proved a lower bound for the
hereditary a discrepancy of a set system F in terms of determinants of square
submatrices of the incidence matrix of F. As shown by an example of Hoffman,
this bound can differ from herdisc(F) by a multiplicative factor of order
almost log n, where n is the size of the ground set of F. We prove that it
never differs by more than O((log n)3/2), assuming |F| bounded by a polynomial
in n. We also prove that if such an F is the union of t systems F_1, . . .,
F_t, each of hereditary discrepancy at most D, then herdisc(F) \leq
O(t^(1/2)(log n)^(3/2) D). For t = 2, this almost answers a question of Sos.
The proof is based on a recent algorithmic result of Bansal, which computes
low-discrepancy colorings using semidefinite programming.
