The maximum sum of the sizes of cross t-intersecting separated families

For a set X and an integer r >= 0, let denote the family of subsets of X that have at <= r (X) (X) most r elements. Two families A subset of and B subset of are cross t-intersecting if |A boolean AND B| >= t for all <= r <= s (X) A is an element of A, B is an element of B. In this paper, we considered the measures of cross t-intersecting families A subset of , ( X)<= r B subset of , then we used this result to obtain the maximum sum of sizes of cross t-intersecting separated <= s families.