PreprintCC BY-NC-ND 4.0arXiv
Forbidden subposet problems in the linear lattice
Abstract
We study weak and strong forbidden subposet problems in the linear lattice Ln(q). We show that the q-analogues of the Bukh--Griggs--Lu conjecture fail for every prime power q. If q≥ 3 and n and d have opposite parity, then Laq(n,Ld(q))=Laq*(n,Ld(q))=Σq(n,d). We also show that Laq(n,Ds)=Laq*(n,Ds)=Σq(n,2) for 2≤ s≤ q and all n, and for s=q+1 when n is odd. For n even, a maximum strong Dq+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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BibTeX
@misc{patkos2026forbidden,
title = {{Forbidden subposet problems in the linear lattice}},
author = {Balázs Patkós and Casey Tompkins},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37748},
eprint = {2609.37748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37748},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Forbidden subposet problems in the linear lattice
AU - Balázs Patkós
AU - Casey Tompkins
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37748
UR - https://arxiv.org/abs/2609.37748
AB - 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≥ 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≤ s≤ 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.
ER -