The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton--Milner family

Abstract: The celebrated Erd\H{o}s-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 Erd\H{o}s-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 Erd\H{o}s-Ko-Rado family nor of the Hilton-Milner family is. For $k\ge 4$, 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 $k\ge 3$ and characterize all extremal families achieving the extremal value.