Extremal problems for suspensions of even cycles

@misc{liu2026extremal,
  title = {{Extremal problems for suspensions of even cycles}},
  author = {Dingyuan Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.22414},
  eprint = {2608.22414},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22414},
}

TY  - UNPB
TI  - Extremal problems for suspensions of even cycles
AU  - Dingyuan Liu
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2608.22414
UR  - https://arxiv.org/abs/2608.22414
AB  - Given an integer k≥2 and a graph F, the k-uniform suspension 𝒮^kF is obtained by adjoining a fixed set of k−2 new vertices to every edge of F. In this paper, we study two extremal problems for suspensions of even cycles. Write K^k_t for the k-uniform clique of order t. Let ex(n,𝒮^kC_2ℓ) and ex(n,K^k_t,𝒮^kC_2ℓ) denote the maximum numbers of edges and copies of K^k_t, respectively, in an 𝒮^kC_2ℓ-free k-uniform hypergraph on n vertices. We prove that, for every k≥2 and infinitely many n, ex(n,K^k_k+1,𝒮^kC_4)=(n^k−1/2)/((k+1)!)+O(n^k−1). This extends a folklore result for k=2 and, as an immediate consequence, yields the asymptotics of ex(n,𝒮^kC_4) for infinitely many n, previously established by Mubayi (for all n). Furthermore, for every k≥2 and ℓ∈{3,5}, we determine the order of magnitude ex(n,𝒮^kC_2ℓ)=Θ(n^k−1+1/ℓ). This generalizes both the classical graph case k=2 and a previous result of Mukherjee for k=ℓ=3. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for ex(n,K^k_k+1,𝒮^kC_4) incorporates a novel block-packing structure, which yields substantially more copies of K^k_k+1 than previously known constructions. For ex(n,𝒮^kC_2ℓ) with ℓ∈{3,5}, we establish a natural k-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.
ER  -