THE MAXIMUM SIZE OF A NON-TRIVIAL INTERSECTING UNIFORM FAMILY THAT IS NOT A SUBFAMILY OF THE HILTON-MILNER FAMILY

The celebrated Erdos-Ko-Rado theorem determines the maximum size of a k-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a k-uniform intersecting family that is not a subfamily of the so-called Erdos-Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting k-uniform family that is neither a subfamily of the Erdos-Ko-Rado family nor of the Hilton-Milner family is. For k >= 4, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. We give a different and simpler proof, based on the shifting method, which allows us to solve all cases k >= 3 and characterize all extremal families achieving the extremal value.