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{\bf Irit Dinur and Ehud Friedgut}
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{\bf Proof of an Intersection Theorem via Graph Homomorphisms}
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Let $0 \leq p \leq 1/2 $ and let $\{0,1\}^n$ be endowed with the
product measure $\mu_p$ defined by
$\mu_p(x)=p^{|x|}(1-p)^{n-|x|}$, where $|x|=\sum x_i$. Let $I
\subseteq \{0,1\}^n$ be an intersecting family, i.e. for every $x,
y \in I$ there exists a coordinate $1 \leq i \leq n$ such that
$x_i=y_i=1$. Then $\mu_p(I) \leq p.$

Our proof uses measure preserving homomorphisms between graphs.

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