A complete solution to the Tokushige measure conjecture and its stability
Abstract
We resolve three conjectures proposed by Tokushige in 2013 about cross t-intersecting families of subsets and integer sequences. For 0<p<1 and ℱ⊆2[n], define the p-biased measure by μp(ℱ)=∑F∈ℱp|F|(1−p)n−|F|. Two families ℱ1,ℱ2⊆2[n] are cross t-intersecting if |F1∩ F2|≥ t for every F1∈ℱ1 and F2∈ℱ2. We prove that, for every n≥ t≥ 2 and 0<p1,p2≤1/(t+1), such a pair satisfies μp1(ℱ1)μp2(ℱ2)≤(p1p2)t. When p1,p2<1/(t+1), equality holds if and only if both families are the same t-star 𝒮T={F⊆[n]:T⊆ F} for some T∈binom[n]t. Together with the previously known case t=1, this completely resolves Tokushige's measure conjecture. We also prove that every pair whose measure product is close to the maximum must be close to a common t-star. More precisely, if p1,p2<1/(t+1) and μp1(ℱ1)μp2(ℱ2)>(1−ε)2(p1p2)t, then there exists T∈binom[n]t such that μpi(ℱimathbin△𝒮T)<Cε for i=1,2, where C depends only on t,p1,p2. This improves Tokushige's conjectured C√ε estimate to Cε. For integer sequences, we prove that if every sequence in ℋ1⊆[m]n agrees with every sequence in ℋ2⊆[m]n in at least t coordinates, then |ℋ1||ℋ2|≤ m2(n−t) for all n≥ t≥1 and m≥ t+1. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection--union product theorem.
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BibTeX
@misc{wu2026complete,
title = {{A complete solution to the Tokushige measure conjecture and its stability}},
author = {Yongjiang Wu and Lihua Feng},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29512},
eprint = {2608.29512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29512},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A complete solution to the Tokushige measure conjecture and its stability
AU - Yongjiang Wu
AU - Lihua Feng
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29512
UR - https://arxiv.org/abs/2608.29512
AB - We resolve three conjectures proposed by Tokushige in 2013 about cross t-intersecting families of subsets and integer sequences. For 0<p<1 and ℱ⊆2^[n], define the p-biased measure by μ_p(ℱ)=∑_F∈ℱp^|F|(1−p)^n−|F|. Two families ℱ_1,ℱ_2⊆2^[n] are cross t-intersecting if |F_1∩ F_2|≥ t for every F_1∈ℱ_1 and F_2∈ℱ_2. We prove that, for every n≥ t≥ 2 and 0<p_1,p_2≤1/(t+1), such a pair satisfies μ_p_1(ℱ_1)μ_p_2(ℱ_2)≤(p_1p_2)^t. When p_1,p_2<1/(t+1), equality holds if and only if both families are the same t-star 𝒮_T={F⊆[n]:T⊆ F} for some T∈binom[n]t. Together with the previously known case t=1, this completely resolves Tokushige's measure conjecture. We also prove that every pair whose measure product is close to the maximum must be close to a common t-star. More precisely, if p_1,p_2<1/(t+1) and μ_p_1(ℱ_1)μ_p_2(ℱ_2)>(1−ε)^2(p_1p_2)^t, then there exists T∈binom[n]t such that μ_p_i(ℱ_imathbin△𝒮_T)<Cε for i=1,2, where C depends only on t,p_1,p_2. This improves Tokushige's conjectured C√ε estimate to Cε. For integer sequences, we prove that if every sequence in ℋ_1⊆[m]^n agrees with every sequence in ℋ_2⊆[m]^n in at least t coordinates, then |ℋ_1||ℋ_2|≤ m^2(n−t) for all n≥ t≥1 and m≥ t+1. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection--union product theorem.
ER -