Almost intersecting families

Let n > k > 1 be integers, [n] = {1, ..., n}. Let F be a family of k-subsets of [n]. The family F is called intersecting if F boolean AND F' not equal empty set for all F, F' is an element of F. It is called almost intersecting if it is not intersecting but to every F is an element of F there is at most one F' is an element of F satisfying F boolean AND F' = empty set. Gerbner et al. proved that if n >= 2k + 2 then vertical bar F vertical bar <= (n-1 k-1) holds for almost intersecting families. Our main result implies the considerably stronger and best possible bound vertical bar F vertical bar <= (n-1 k-1) - (n-k-1 k-1) + 2 for n > (2 + o(1))k, k >= 3.