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Turán numbers of triangular pyramids
Abstract
Let TPk denote the triangular pyramid with k layers. For a graph family ℱ, let ex(n,ℱ) be the maximum number of edges in an n-vertex graph containing no member of ℱ as a subgraph. For a graph H, let cJ(H) be the family of subgraph-minimal graphs among all such graphs H−I, where I⊆ V(H) is an independent set. For every integer k≥4 and all sufficiently large n, we prove that ex(n,TPk)=max0≤ m≤ n{m(n−m)+ex(m,cJ(TPk))}+Ok(n)=⌊ n2/4⌋+Θk(n4/3). Moreover, for k=4, we characterize all extremal graphs and show that ex(n,TP4)=max0≤ m≤ n{m(n−m)+ex(m,cJ(TP4))} for sufficiently large n.
Related
BibTeX
@misc{xiao2026turán,
title = {{Turán numbers of triangular pyramids}},
author = {Chuanqi Xiao and Long-Tu Yuan and Xiaowen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18493},
eprint = {2610.18493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18493},
}
RIS
TY - UNPB
TI - Turán numbers of triangular pyramids
AU - Chuanqi Xiao
AU - Long-Tu Yuan
AU - Xiaowen Zhang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18493
UR - https://arxiv.org/abs/2610.18493
AB - Let TP_k denote the triangular pyramid with k layers. For a graph family ℱ, let ex(n,ℱ) be the maximum number of edges in an n-vertex graph containing no member of ℱ as a subgraph. For a graph H, let cJ(H) be the family of subgraph-minimal graphs among all such graphs H−I, where I⊆ V(H) is an independent set. For every integer k≥4 and all sufficiently large n, we prove that ex(n,TP_k)=max_0≤ m≤ n{m(n−m)+ex(m,cJ(TP_k))}+O_k(n)=⌊ n^2/4⌋+Θ_k(n^4/3). Moreover, for k=4, we characterize all extremal graphs and show that ex(n,TP_4)=max_0≤ m≤ n{m(n−m)+ex(m,cJ(TP_4))} for sufficiently large n.
ER -