Let \(\mathcal {A}=\{A_{1},\ldots ,A_{p}\}\) and \(\mathcal {B}=\{B_{1},\ldots ,B_{q}\}\) be two families of subsets of [n] such that for every \(i\in [p]\) and \(j\in [q]\), \(|A_{i}\cap B_{j}|= \frac{c}{d}|B_{j}|\), where \(\frac{c}{d}\in [0,1]\) is an irreducible fraction. We call such families \(\frac{c}{d}\)-cross intersecting families. In this paper, we find a tight upper bound for the product \(|\mathcal {A}||\mathcal {B}|\) and characterize the cases when this bound is achieved for \(\frac{c}{d}=\frac{1}{2}\). Also, we find a tight upper bound on \(|\mathcal {A}||\mathcal {B}|\) when \(\mathcal {B}\) is k-uniform and characterize, for all \(\frac{c}{d}\), the cases when this bound is achieved.

令\（\ mathcal {A} = \ {A_ {1}，\ ldots，A_ {p} \} \）和\（\ mathcal {B} = \ {B_ {1}，\ ldots，B_ {q} \} \）是[ n ]的两个子集，因此对于每个\ [i \ in [p] \）和\（j \ in [q] \），\（| A_ {i} \ cap B_ { j} | = \ frac {c} {d} | B_ {j} | \），其中\（\ [0,1] \中的\ frac {c} {d} \）是不可约分数。我们称这类家庭为\（\ frac {c} {d} \）交叉的家庭。在本文中，我们找到了\（| \ mathcal {A} || \\ mathcal {B} | \）的紧上限，并刻画了\（\ frac {c} {d } = \ frac {1} {2} \）。此外，我们发现\（| \ mathcal {A} ||| \ mathcal {B} | \）当\（\ mathcal {B} \）为k时，对于所有\（\ frac {c} {d} \），其特征都是，实现此限制的情况。