ON CROSS-INTERSECTING FAMILIES

Let n greater-than-or-equal-to t greater-than-or-equal-to 1 be integers. Let F, G be families of subsets of the n-element set X. They are called cross t-intersecting if \F and G\ greater-than-or-equal-to t holds for all F is-an-element-of F and G is-an-element-of G. If F = G then F is called t-intersecting. Let m(n, t) denote the maximum possible cardinality of a t-intersecting family, Our main result says that if F, G are cross s-intersecting with Absolute value of G less-than-or-equal-to Absolute value of F less-than-or-equal-to m(n, t), t less-than-or-equal-to s, then Absolute value of F + Absolute value of G less-than-or-equal-to m(n,t) + m(n, 2s - t) holds and this is best possible.