Forbidden sparse intersections

Abstract Let n be a positive integer, let $0<p\leqslant p'\leqslant \frac 12$ , and let $\ell \leqslant pn$ be a nonnegative integer. We prove that if $\mathcal {F},\mathcal {G}\subseteq \{0,1\}^n$ are two families whose cross intersections forbid $\ell $ —that is, they satisfy $|A\cap B|\neq \ell $ for every $A\in \mathcal {F}$ and every $B\in \mathcal {G}$ – then, setting $t:= \min \{\ell ,pn-\ell \}$ , we have the subgaussian bound $$\begin{align*}\mu_p(\mathcal{F})\, \mu_{p'}(\mathcal{G}) \leqslant 2\exp\Big( - \frac{t^2}{58^2\,pn}\Big), \end{align*}$$ where $\mu _p$ and $\mu _{p'}$ denote the p -biased and $p'$ -biased measures on $\{0,1\}^n$ , respectively.