Turán Numbers of Linear Stars and Expanded Trees
Abstract
For an r-uniform hypergraph ℋ, let exr(n,ℋ) denote the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of ℋ. The r-uniform expansion T(r) of a tree T is obtained by adding r−2 new vertices to each edge, with the added sets pairwise disjoint and disjoint from V(T). In particular, the expansion of K1,k is the linear star ℒkr, whose k edges intersect pairwise in a single common vertex. Chung and Frankl (1987) determined ex3(n,ℒk3) and all extremal hypergraphs for every k≥3 and sufficiently large n. We extend the odd-k part of their result to every r≥5, determining the exact Turán number and the unique extremal hypergraph up to isomorphism for sufficiently large n. More generally, we extend this result to expansions of trees for which the minimum degree among the vertices in the smaller color class is odd and at least three. For every r≥5 and sufficiently large n, we establish lower and upper bounds on their Turán numbers. When no two vertices attaining this minimum degree in that color class have a common neighbor, we determine the exact Turán number and the unique extremal hypergraph up to isomorphism. The proofs combine the delta-system method with stability arguments and a detailed analysis of extremal constructions.
Related
BibTeX
@misc{hou2026turán,
title = {{Turán Numbers of Linear Stars and Expanded Trees}},
author = {Xinmin Hou and Anyao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15447},
eprint = {2610.15447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15447},
}
RIS
TY - UNPB
TI - Turán Numbers of Linear Stars and Expanded Trees
AU - Xinmin Hou
AU - Anyao Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15447
UR - https://arxiv.org/abs/2610.15447
AB - For an r-uniform hypergraph ℋ, let ex_r(n,ℋ) denote the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of ℋ. The r-uniform expansion T^(r) of a tree T is obtained by adding r−2 new vertices to each edge, with the added sets pairwise disjoint and disjoint from V(T). In particular, the expansion of K_1,k is the linear star ℒ_k^r, whose k edges intersect pairwise in a single common vertex. Chung and Frankl (1987) determined ex_3(n,ℒ_k^3) and all extremal hypergraphs for every k≥3 and sufficiently large n. We extend the odd-k part of their result to every r≥5, determining the exact Turán number and the unique extremal hypergraph up to isomorphism for sufficiently large n. More generally, we extend this result to expansions of trees for which the minimum degree among the vertices in the smaller color class is odd and at least three. For every r≥5 and sufficiently large n, we establish lower and upper bounds on their Turán numbers. When no two vertices attaining this minimum degree in that color class have a common neighbor, we determine the exact Turán number and the unique extremal hypergraph up to isomorphism. The proofs combine the delta-system method with stability arguments and a detailed analysis of extremal constructions.
ER -