Tokushige's conjecture on measures of cross $t$-intersecting families
Yongjiang Wu, Yongtao Li, Zhiyi Liu, Lihua Feng
TL;DR
The paper addresses Tokushige's conjectures on cross $t$-intersecting families in both the measure and integer-sequence settings. It proves the product-measure bound $\mu_{p_1}(\mathcal{F}_1)\mu_{p_2}(\\mathcal{F}_2)\le (p_1p_2)^t$ for $n\ge t\ge 3$ and $p_1,p_2\in\left(0,\frac{1}{t+1}\right)$, resolving the conjecture in this regime; it also establishes $|\mathcal{H}_1||\mathcal{H}_2|\le (m^{n-t})^2$ for cross $t$-intersecting $[m]^n$-families with $m>t+1$, confirming the corresponding Tokushige conjecture. As applications, the authors extend the IU-Theorem via $(t_1,\dots,t_m)$-intersecting frameworks and derive Katona-type measure bounds, including a Katona-family extremal result for $p\ge\tfrac{1}{2}$. The methods combine algebraic cross-intersection bounds (notably the He–Li–Wu–Zhang result), probabilistic concentration, and extended shifting techniques to transfer bounds between set families and sequence-based families. These results advance the understanding of extremal cross-intersection problems in both discrete and product-measure settings and link classical extremal theory with modern probabilistic methods.
Abstract
We investigate the product measure version (also known as the $p$-weight version) of intersection problems in extremal combinatorics. Firstly, we prove that for any \(n \geq t \geq 3\) and \(p_1, p_2 \in (0, \frac{1}{t+1})\), if \(\mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}\) are cross \(t\)-intersecting families, then $μ_{p_1}(\mathcal{F}_1)μ_{p_2}(\mathcal{F}_2)\le (p_1p_2)^t$. This resolves a conjecture of Tokushige for \(t \geq 3\). Secondly, we study the intersection problems for integer sequences and prove that if $\mathcal{H}_1, \mathcal{H}_2 \subseteq [m]^{n}$ are cross $t$-intersecting with \(m > t+1\), then $|\mathcal{H}_1|| \mathcal{H}_2|\leq (m^{n-t})^2$. This confirms another conjecture of Tokushige for \(m > t+1\). As an application, we strengthen a recent theorem of Frankl and Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if \(p \geq \frac{1}{2}\) and $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \left\{μ_{p}(\mathcal{F}_1),μ_{p}(\mathcal{F}_2)\right\} \leq μ_{p}(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede and Katona.