On forbidden poset problems in the linear lattice

TL;DR

This paper determines the maximum size of certain forbidden poset families in the lattice of vector subspaces, generalizing previous results and proving a new LYM-type lemma for the linear lattice.

Contribution

It provides the first comprehensive bounds for $ ext{V}_k, ext{Lambda}_l$-free families in the linear lattice and proves a conjecture of Shahriari and Yu.

Findings

Maximum size bounds for forbidden poset families established

General LYM-type lemma for linear lattice proved

Results extend previous work by Shahriari and Yu

Abstract

In this note, we determine the maximum size of a $\{V_{k}, \Lambda_{l}\}$-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.