A sharp $p$-biased product bound for $r$-cross-intersecting families

We prove a sharp product theorem for $r$-cross-intersecting families in the $p$-biased measure. If $r\ge 2$, $0\le p\le \frac{r-1}{r}$, and ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_r\subseteq 2^{[n]}$ are $r$-cross-intersecting, then $$\prod _{i=1}^r{\mu }_p({\mathcal{F}}_i)\le p^r\mathrm{.}$$ The bound is attained by a common $1$-star, and the range of $p$ is best possible. In particular, this proves the equal-bias case of a conjecture of Frankl and Tokushige and, for $r=3$, confirms a conjecture of Tokushige. We also prove a stability theorem: for $r\ge 3$, every near-extremal $r$-tuple is close, in $p$-biased measure, to a common $1$-star, with an optimal linear dependence on the product deficit. The extremal proof uses a coordinatewise coupling at the critical bias together with an isoperimetric inequality for increasing families. The stability proof uses dual families and random ordered partitions to obtain Fourier concentration, then applies biased Friedgut–Kalai–Naor theorem to force the star structure.