Improved Upper Bounds Concerning the Erdős-Ko-Rado Theorem

A family ℱ of k -element sets of an n -set is called t -intersecting if any two of its members overlap in at least t -elements. The Erdős-Ko-Rado Theorem gives a best possible upper bound for such a family if n ≥ n 0 ( k, t ). One of the most exciting open cases is when t = 2, n = 2 k . The present paper gives an essential improvement on the upper bound for this case. The proofs use linear algebra and yield more general results.