Abstract
In this note, we determine the maximum size of a $\{\mathrm{V}_{k}, \Lambda_{l}\}$-free family in the lattice of vector subspaces of a finite vector space both in the non-induced case as well as the induced case, for a large range of parameters $k$ and $l$. These results generalize earlier work by Shahriari and Yu. We also prove a general LYM-type lemma for the linear lattice which resolves a conjecture of Shahriari and Yu.
Highlights
Given partially ordered sets P and Q, we say that P is a subposet of Q if there exists an injection φ : P → Q such that x P y implies φ(x) Q φ(y)
In the vector space setting, Shahriari and Yu [13] proved a version of Theorem 9 holds
Equality occurs only for a family consisting of the union of two consecutive levels in the linear lattice of maximum size
Summary
Given partially ordered sets (posets) P and Q, we say that P is a subposet of Q if there exists an injection φ : P → Q such that x P y implies φ(x) Q φ(y). In the vector space setting, Shahriari and Yu [13] proved a version of Theorem 9 holds Equality occurs only for a family consisting of the union of two consecutive levels in the linear lattice of maximum size. They posed a conjecture for the case when {Yk, Yk} is forbidden. Burcsi and Nagy [1] and Grosz, Methuku and Tompkins [8] proved the following theorems for any poset P (another result in this direction was obtain by Chen and Li [2]). We note that a recent manuscript of Gerbner [6] independently initiates a general study of LYM-type properties of the linear lattice and implies some similar results
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