On the Frankl--Tokushige conjecture and almost complete r-cross t-intersection theorems for vector spaces
Abstract
Let r≥3 and k1≥ k2≥⋯≥ kr≥ t. Let ℱ1,ℱ2,…,ℱr be families of subspaces, of respective dimensions k1,k2,…,kr, in an n-dimensional vector space over the finite field 𝔽q. The r families are called r-cross t-intersecting if dim (F1 ∩ F2 ∩ ⋯ ∩ Fr) ≥ t for all Fi ∈ ℱi, i = 1,2,…,r. In 2016, Frankl and Tokushige conjectured that ∏i=1r|ℱi|≤∏i=1rn−1brack ki−1 for t=1 and n≥ rk1/(r−1). The appealing conjecture suggests establishing intersection theorems for n∼ ck1 with c=c(r)∈(1,2), a direction that has long been challenging. In this paper, we overcome this barrier by proving that ∏i=1r|ℱi|≤∏i=1rn−tbrack ki−t for all t≥1 and n≥ rk1/(r−1)+C(t,r), where C(t,r)=rt/(r−1)+1. This proves the Frankl--Tokushige conjecture except for at most three values of n, and establishes an Erdős--Ko--Rado type theorem for almost all values of parameters. Furthermore, we characterize all extremal configurations. Our proof is purely combinatorial and based on the t-cover method, with several essential refinements. We also obtain almost complete intersection theorems for r-wise t-intersecting families and non-trivial r-cross t-intersecting families.
Related
- Intersecting families with large covering number - On a conjecture of Frankl, Ota, and Tokushige
- A complete solution to the Tokushige measure conjecture and its stability
- Confirmation of the Daykin-Frankl Conjecture
- The asymptotic Erd{ő}s--Frankl--Füredi cover-free conjecture
- A proof of the strong Erdős--Frankl--Füredi cover-free conjecture
BibTeX
@misc{li2026frankltokushige,
title = {{On the Frankl--Tokushige conjecture and almost complete $r$-cross $t$-intersection theorems for vector spaces}},
author = {Yao Li and Benjian Lv and Jie Wen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05848},
eprint = {2609.05848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05848},
}
RIS
TY - UNPB
TI - On the Frankl--Tokushige conjecture and almost complete r-cross t-intersection theorems for vector spaces
AU - Yao Li
AU - Benjian Lv
AU - Jie Wen
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05848
UR - https://arxiv.org/abs/2609.05848
AB - Let r≥3 and k_1≥ k_2≥⋯≥ k_r≥ t. Let ℱ_1,ℱ_2,…,ℱ_r be families of subspaces, of respective dimensions k_1,k_2,…,k_r, in an n-dimensional vector space over the finite field 𝔽_q. The r families are called r-cross t-intersecting if dim (F_1 ∩ F_2 ∩ ⋯ ∩ F_r) ≥ t for all F_i ∈ ℱ_i, i = 1,2,…,r. In 2016, Frankl and Tokushige conjectured that ∏_i=1^r|ℱ_i|≤∏_i=1^rn−1brack k_i−1 for t=1 and n≥ rk_1/(r−1). The appealing conjecture suggests establishing intersection theorems for n∼ ck_1 with c=c(r)∈(1,2), a direction that has long been challenging. In this paper, we overcome this barrier by proving that ∏_i=1^r|ℱ_i|≤∏_i=1^rn−tbrack k_i−t for all t≥1 and n≥ rk_1/(r−1)+C(t,r), where C(t,r)=rt/(r−1)+1. This proves the Frankl--Tokushige conjecture except for at most three values of n, and establishes an Erdős--Ko--Rado type theorem for almost all values of parameters. Furthermore, we characterize all extremal configurations. Our proof is purely combinatorial and based on the t-cover method, with several essential refinements. We also obtain almost complete intersection theorems for r-wise t-intersecting families and non-trivial r-cross t-intersecting families.
ER -