A size-sensitive inequality for cross-intersecting families

Two families A and B, of k-subsets of an n-set are called cross intersecting if A boolean AND B not equal phi for all A,B epsilon b. Strengthening the classical ErclOs-Ko-Rado theorem, Pyber proved that vertical bar A vertical bar vertical bar B vertical bar 2k. In the present paper we sharpen this inequality. We prove that assuming vertical bar B vertical bar >= ((n - 1 k - 1) - (n -1 k -1)) for some 3