Forbidden sparse intersections
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 sati...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100674/type/journal_article |
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| Summary: | 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
$$\begin{align*}\mu_p(\mathcal{F})\, \mu_{p'}(\mathcal{G}) \leqslant 2\exp\Big( - \frac{t^2}{58^2\,pn}\Big), \end{align*}$$
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| ISSN: | 2050-5094 |