Tokushige测度猜想及其稳定性的完整解决方案
A complete solution to the Tokushige measure conjecture and its stability
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中文总结 AI 辅助
该研究完全解决了Tokushige关于交叉t-相交族的测度猜想，证明了相关不等式及等号成立条件，得到了更强的稳定性结果，并将结论推广至整数序列情形，改进了已有估计。
中文摘要 AI 辅助
我们解决了Tokushige在2013年提出的关于子集和整数序列的交叉t-相交族的三个猜想。对于0<p<1且F⊆2^{[n]}，将p-偏倚测度定义为μ_p(F)=∑_{F∈F}p^{|F|}(1-p)^{n-|F|}。若对所有F₁∈F₁、F₂∈F₂都有|F₁∩F₂|≥t，则两个族F₁,F₂⊆2^{[n]}是交叉t-相交的。我们证明，对所有n≥t≥2且0<p₁,p₂≤1/(t+1)，这类族对满足μ_{p₁}(F₁)μ_{p₂}(F₂)≤(p₁p₂)^t。当p₁,p₂<1/(t+1)时，等号成立当且仅当两个族是同一个t-星S_T={F⊆[n]:T⊆F}，其中T∈binom([n],t)。结合此前已知的t=1情形，这完全解决了Tokushige的测度猜想。我们还证明，所有测度乘积接近最大值的族对都必然接近某个公共t-星。更确切地说，若p₁,p₂<1/(t+1)且μ_{p₁}(F₁)μ_{p₂}(F₂)>(1-ε)²(p₁p₂)^t，则存在T∈binom([n],t)，使得对i=1,2有μ_{p_i}(F_i△S_T)<Cε，其中C仅依赖于t,p₁,p₂。这将Tokushige猜想的C√ε估计改进为Cε。对于整数序列，我们证明若H₁⊆[m]^n中的每个序列与H₂⊆[m]^n中的每个序列至少在t个坐标上一致，则对所有n≥t≥1且m≥t+1，有|H₁||H₂|≤m^{2(n-t)}。我们进一步得到了更一般的结果，即对每个可能的取值施加单独的一致性要求。这推广了Frankl和Kupavskii的一个定理，并还原了他们此前的交叉相交-并乘积定理。
英文摘要
We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0<p<1$ and $\mathcal F\subseteq2^{[n]}$, define the $p$-biased measure by $μ_p(\mathcal F)=\sum_{F\in\mathcal F}p^{|F|}(1-p)^{n-|F|}$. Two families $\mathcal F_1,\mathcal F_2\subseteq2^{[n]}$ are cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for every $F_1\in\mathcal F_1$ and $F_2\in\mathcal F_2$. We prove that, for every $n\geq t\geq 2$ and $0<p_1,p_2\leq1/(t+1)$, such a pair satisfies $μ_{p_1}(\mathcal F_1)μ_{p_2}(\mathcal F_2)\leq(p_1p_2)^t$. When $p_1,p_2<1/(t+1)$, equality holds if and only if both families are the same $t$-star $\mathcal S_T=\{F\subseteq[n]:T\subseteq F\}$ for some $T\in\binom{[n]}{t}$. 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 $μ_{p_1}(\mathcal F_1)μ_{p_2}(\mathcal F_2)>(1-\varepsilon)^2(p_1p_2)^t$, then there exists $T\in\binom{[n]}{t}$ such that $μ_{p_i}(\mathcal F_i\mathbin{\triangle}\mathcal S_T)<C\varepsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige's conjectured $C\sqrt{\varepsilon}$ estimate to $C\varepsilon$. For integer sequences, we prove that if every sequence in $\mathcal H_1\subseteq[m]^n$ agrees with every sequence in $\mathcal H_2\subseteq[m]^n$ in at least $t$ coordinates, then $|\mathcal H_1||\mathcal H_2|\leq m^{2(n-t)}$ for all $n\geq t\geq1$ and $m\geq t+1$. 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.
发表机构
- School of Mathematics and Statistics, HNP-LAMA, Central South University（中南大学数学与统计学院，HNP-LAMA）
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