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Upper Bounds on the Turán Density of Hypergraphs Associated with the Projective Plane over a Finite Field
Abstract
For a prime power q≥2 and an integer m≥2, let PG(m,q) denote the m−dimensional projective hypergraph over the finite field 𝔽q. Employing the Lp−method and using the incidence structure, we establish the Turán density of PG(m,q) π(PG(m,q))≤ (1−1/(q+1))1/q & if m=2, (1−1/((q+1)|E(PG(m−1,q))|))1/(1+qm−1) & if m≥3. The previously known upper bound on its Turán density was 1−binomqmq−1. As a consequence of our estimate, this upper bound is improved for each prime power q≥3 and for each dimension index m≥2. For a positive integer t, let Bt[PG(2,q)] be the (q+1)−graph formed by taking t copies of the PG(2,q) that shares a common edge. Using the Lp−method, we establish that the upper bound of the Turán density of Bt[PG(2,q)] is (1−1/(q+1))1/qt.
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BibTeX
@misc{dash2026upper,
title = {{Upper Bounds on the Turán Density of Hypergraphs Associated with the Projective Plane over a Finite Field}},
author = {Subhankar Dash and Kaushik Majumder},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01084},
eprint = {2610.01084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01084},
}
RIS
TY - UNPB
TI - Upper Bounds on the Turán Density of Hypergraphs Associated with the Projective Plane over a Finite Field
AU - Subhankar Dash
AU - Kaushik Majumder
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01084
UR - https://arxiv.org/abs/2610.01084
AB - For a prime power q≥2 and an integer m≥2, let PG(m,q) denote the m−dimensional projective hypergraph over the finite field 𝔽_q. Employing the L^p−method and using the incidence structure, we establish the Turán density of PG(m,q) π(PG(m,q))≤ (1−1/(q+1))^1/q & if m=2, (1−1/((q+1)|E(PG(m−1,q))|))^1/(1+q^m−1) & if m≥3. The previously known upper bound on its Turán density was 1−binomq^mq^−1. As a consequence of our estimate, this upper bound is improved for each prime power q≥3 and for each dimension index m≥2. For a positive integer t, let B_t[PG(2,q)] be the (q+1)−graph formed by taking t copies of the PG(2,q) that shares a common edge. Using the L^p−method, we establish that the upper bound of the Turán density of B_t[PG(2,q)] is (1−1/(q+1))^1/qt.
ER -