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The celebrated Erd\H{o}s-Ko-Rado theorem determines the maximum size of a$k$-uniform intersecting family. The Hilton-Milner theorem determines themaximum size of a $k$-uniform intersecting family that is not a subfamily ofthe so-called Erd\H{o}s-Ko-Rado family. In turn, it is natural to ask what themaximum size of an intersecting $k$-uniform family that is neither a subfamilyof the Erd\H{o}s-Ko-Rado family nor of the Hilton-Milner family is. For $k\ge4$, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. Wegive a different and simpler proof, based on the shifting method, which allowsus to solve all cases $k\ge 3$ and characterize all extremal families achievingthe extremal value.

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