Geometrical Solution of an Intersection Problem for Two Hypergraphs

Let A1, A2, ..., Am, be at most a and let B1, ..., Bm, be at most b-element sets and let t be a non-negative integer with the following property |Ai ∩ Bi| ≤ t and |Ai ∩ Bi| >t for i ≠ j. Then m ⩽ ( a + b - 2 t a − t ) . The proof uses Lovasz's geometrical method and leads to several open problems.