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∩B≠0̸ for all A∈A,B∈B. Strengthening the classical Erdős-Ko-Rado theorem, Pyber proved that |A||B|≤n−1k−12 holds for n≥2k. In the present paper we sharpen this inequality. We prove that assuming |B|≥n−1k−1+n−ik−i+1 for some 3≤i≤k+1 the stronger inequality |A||B|≤(n−1k−1+n−ik−i+1)(n−1k−1−n−ik−1) holds. These inequalities are best possible. We also present a new short proof of Pyber's inequality and a short computation-free proof of an inequality due to Frankl and Tokushige (1992).