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arXiv · 2609.37748

Forbidden subposet problems in the linear lattice

Balázs Patkós·Casey Tompkins
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Abstract

We study weak and strong forbidden subposet problems in the linear lattice $L_n(q)$. We show that the $q$-analogues of the Bukh--Griggs--Lu conjecture fail for every prime power $q$. If $q\ge 3$ and $n$ and $d$ have opposite parity, then $La_q(n,L_d(q))=La_q^*(n,L_d(q))=Σ_q(n,d)$. We also show that $La_q(n,D_s)=La_q^*(n,D_s)=Σ_q(n,2)$ for $2\le s\le q$ and all $n$, and for $s=q+1$ when $n$ is odd. For $n$ even, a maximum strong $D_{q+1}$-free family can be contained from the three middle layers. The case $s=2$ settles the $q$-analog of the diamond conjecture in the affirmative.

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Balázs Patkós, Casey Tompkins. 2026-09-29. Forbidden subposet problems in the linear lattice. https://arxiv.org/abs/2609.37748

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