Let $H$ and $K$ be finitely generated subgroups of a free group $F$, and suppose that $H$ has finite index in $F$. Is it true that $rank(H \cap K)1 \leq (rank(H)1)(rank(K)1)$?

Yes, this statement is true and for a long time was known as the Hanna Neumann Conjecture. It was proved in 2011 by Igor Mineyev. 

