PreprintCC BY 4.0arXiv
Counting t-wise L-intersecting cliques with prescribed intersection sizes
Abstract
Let r,t≥ 3 be integers and L={ℓ1,ℓ2,…,ℓs}⊆ [0,r−1] a fixed set of integers with |L|≠ r and ℓ1<ℓ2<⋯<ℓs. For each integer n, let Ψr(n,L,t) be the maximum number of r-cliques in an n-vertex graph whose r-cliques, viewed as a family of r-subsets of the vertex set, form a t-wise L-intersecting family. In this paper, we prove that Ψr(n,L,t)=o(n|L|) when the sequence ℓ1,ℓ2,…,ℓs,r does not form an arithmetic progression, and we give an asymptotic formula for Ψr(n,L,t) when this sequence does form an arithmetic progression. When t=2, our results are exactly Helliar and Liu's results.
Related
- Turán number of three disjoint cliques of arbitrary sizes
- Normal approximation for subgraph counts in random intersection graphs: the case of cliques
- Some advances on stretching operations applied to cliques of edge intersection graphs of paths in trees
- Sizes of cross-intersecting families
- The sum of sizes of nontrivial cross-intersecting families
BibTeX
@misc{zhan2026counting,
title = {{Counting $t$-wise $L$-intersecting cliques with prescribed intersection sizes}},
author = {Yiyan Zhan and Yichen Wang and Mei Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22393},
eprint = {2609.22393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22393},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting t-wise L-intersecting cliques with prescribed intersection sizes
AU - Yiyan Zhan
AU - Yichen Wang
AU - Mei Lu
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22393
UR - https://arxiv.org/abs/2609.22393
AB - Let r,t≥ 3 be integers and L={ℓ_1,ℓ_2,…,ℓ_s}⊆ [0,r−1] a fixed set of integers with |L|≠ r and ℓ_1<ℓ_2<⋯<ℓ_s. For each integer n, let Ψ_r(n,L,t) be the maximum number of r-cliques in an n-vertex graph whose r-cliques, viewed as a family of r-subsets of the vertex set, form a t-wise L-intersecting family. In this paper, we prove that Ψ_r(n,L,t)=o(n^|L|) when the sequence ℓ_1,ℓ_2,…,ℓ_s,r does not form an arithmetic progression, and we give an asymptotic formula for Ψ_r(n,L,t) when this sequence does form an arithmetic progression. When t=2, our results are exactly Helliar and Liu's results.
ER -