Weighted cross-intersecting families

In this paper we investigate weighted cross-intersecting families: if α,β>0α,β>0 are given constants, we want to find the maximum of α|A|+β|B|α|A|+β|B| for A,BA,B uniform cross-intersecting families. We determine the maximum sum, even if we have restrictions of the size of AA.As corollaries, we will obtain some new bounds on the shadows and the shades of uniform families. We give direct proofs for these bounds, as well, and show that the theorems for cross-intersecting families also follow from these results.Finally, we will generalize the LYM inequality not only for cross-intersecting families, but also for arbitrary Sperner families.