A Hilton-Milner theorem for exterior algebras

Recent work of Scott and Wilmer and of Woodroofe extends the Erdős-Ko-Rado theorem from set systems to subspaces of k-forms in an exterior algebra. We prove an extension of the Hilton-Milner theorem to the exterior algebra setting, answering in a strong way a question asked by these authors.

💡 Research Summary

The paper establishes a Hilton‑Milner type theorem for subspaces of k‑forms in an exterior algebra, thereby answering a question raised by Scott‑Wilmer and Woodroofe. The classical Erdős‑Ko‑Rado (EKR) theorem gives an upper bound on the size of a pairwise‑intersecting family of k‑subsets of an n‑element set, with equality only for families consisting of all k‑sets containing a fixed element. Scott and Wilmer, and later Woodroofe, lifted this result to the setting of exterior algebras: if L⊆∧^k V is a self‑annihilating subspace (i.e., x∧y=0 for all x,y∈L) then dim L≤C(n−1,k−1). This is a direct analogue of the EKR bound.

The authors now consider the “non‑trivial’’ case, mirroring the Hilton‑Milner theorem. A self‑annihilating subspace L is called non‑trivial if there is no 1‑form v∈∧^1 V that annihilates the whole of L (equivalently, L is not contained in the kernel of any wedge with a fixed vector). Their main result (Theorem 1.5) shows that for such a non‑trivial L, \