Linear bounds on characteristic polynomials of matroids

Let chi(t)= a(0)t(n) - a(1)t(n-1) + ... + (-1)(r)a(r)t(n-r) be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, ..., r and real number x >= k - r - 1, we obtain a linear bound of the coefficient sequence, that is (r + x k) <= Sigma(k)(i=0) a(i) (x k - i) <= (m + x k), where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney's sign-alternating theorem, Meredith's upper bound theorem, and Dowling and Wilson's lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.