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

Let f(x 1 ; : : : ; x n ) = _ I2F ^ i2I x i and g(x 1 ; : : : ; x n ) = _ I2G ^ i2I x i 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 x i ; i = 1; : : : ; n, let i = #fI 2 F ji 2 Ig=jF j and i = #fI 2 Gji 2 Ig=jGj be the frequencies with which x i occurs in f and g. It is easily seen that maxf 1 ; 1 ; : : : ; n ; n g 1= log(jF j+ jGj):We give examples of arbitrarily large F and G for which the above bound is tight up to a factor of 2.