On the frequency of the most frequently occurring variable in dual monotone DNFs

Let f(x1,....., xn) = VIεF^iεIxi and g(x1,.....,xn) = VIεF^iεIxi be a pair of dual monotone irredundant disjunctive normal forms, where F and G are the sets of the prime implicants of f and g, respectively. For a variable xi, i = 1,......, n, let i= #{I ε F|i ε I}/|F| and i = #{I ε G|i ε I}/|G| be the frequencies with which xi occurs in f and g. It is easily seen that maxf{u1, v1,......un,vn}>= 1/ log(|F|+|G|): We give examples of arbitrarily large F and G for which the above bound is tight up to a factor of 2.Technical report LCSR-TR-25