A complete solution to the Tokushige measure conjecture and its stability

We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0<p<1$ and $\mathcal{F}\subseteq 2^{[n]}$, define the $p$-biased measure by ${\mu }_p(\mathcal{F})={\sum }_{F\in \mathcal{F}}{p}^{\mathrm{\mid }F\mathrm{\mid }}(1-p{)}^{n-\mathrm{\mid }F\mathrm{\mid }}$. Two families ${\mathcal{F}}_1,{\mathcal{F}}_2\subseteq 2^{[n]}$ are cross $t$-intersecting if $\mathrm{\mid }{F}_1\cap F_2\mathrm{\mid }\ge t$ for every $F_1\in {\mathcal{F}}_1$ and $F_2\in {\mathcal{F}}_2$. We prove that, for every $n\ge t\ge 2$ and $0<p_1,p_2\le 1\mathrm{/}(t+1)$, such a pair satisfies ${\mu }_{p_1}({\mathcal{F}}_1){\mu }_{p_2}({\mathcal{F}}_2)\le (p_1{p}_2{)}^t$. When $p_1,p_2<1\mathrm{/}(t+1)$, equality holds if and only if both families are the same $t$-star ${\mathcal{S}}_T=\{F\subseteq [n]:T\subseteq F\}$ for some $T\in \left(\genfrac{}{}{0px}{}{[n]}{t}\right)$. 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 $p_1,p_2<1\mathrm{/}(t+1)$ and ${\mu }_{p_1}({\mathcal{F}}_1){\mu }_{p_2}({\mathcal{F}}_2)>(1-\epsilon {)}^2(p_1{p}_2{)}^t$, then there exists $T\in \left(\genfrac{}{}{0px}{}{[n]}{t}\right)$ such that ${\mu }_{p_i}({\mathcal{F}}_i\mathrm{△}{\mathcal{S}}_T)<C\epsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige’s conjectured $C\sqrt{\epsilon }$ estimate to $C\epsilon$. 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 $\mathrm{\mid }{\mathcal{H}}_1\mathrm{\mid }\mathrm{\mid }{\mathcal{H}}_2\mathrm{\mid }\le m^{2(n-t)}$ for all $n\ge t\ge 1$ and $m\ge t+1$. 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.