Avoiding Brooms, Forks, and Butterflies in the Linear Lattices

Abstract

Let n be a positive integer, q a power of a prime, and $\mathcal {L}_{n}({q})$ the poset of subspaces of an n-dimensional vector space over a field with q elements. This poset is a normalized matching poset and the set of subspaces of dimension ⌊n/2⌋ or those of dimension ⌈n/2⌉ are the only maximum-sized antichains in this poset. Strengthening this well-known and celebrated result, we show that, except in the case of $\mathcal {L}_{3}({2})$, these same collections of subspaces are the only maximum-sized families in $\mathcal {L}_{n}({q})$ that avoid both a ∧ and a ∨ as a subposet. We generalize some of the results to brooms and forks, and we also show that the union of the set of subspaces of dimension k and k + 1, for k = ⌊n/2⌋ or k = ⌈n/2⌉ − 1, are the only maximum-sized families in $\mathcal {L}_{n}({q})$ that avoid a butterfly (definitions below).