FORBIDDEN CONFIGURATIONS, DISCREPANCY AND DETERMINANTS

An m × n matrix A is said to have hereditary discrepancy d if the maximum over submatrices B of A of the minimum over (−1/2, +1/2)-vectors x of ‖Bx‖∞. If A is integral and has no repeated rows then we show that m is bounded by a polynomial in n for fixed d. This improves a result of Lovász and Vesztergombi for fixed d. Let A be an m × n matrix with no repeated rows and with each square submatrix having determinant {0, ±1, ±2,..., ±k} . Then m is bounded by a polynomial in n for fixed k. This extends a result of Heller for totally unimodular matrices. (A result of Kung combined with results of J. Lee provides a much better bound.) Both bounds follow from repeated applications of the forbidden configuration theorem of Sauer, Perles and Shelah, which states that an m × n (0, 1)-matrix with no repeated rows and no 2kxk submatrix of all (0, 1)-rows on k columns has m at most ( n k−1)+ ( n k−2)+⋯+ ( n0). © 1990, Academic Press Limited. All rights reserved.