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A set of sets is called a  family . Two families   A   and   B   are said to be  cross-intersecting  if each set in   A   intersects each set in   B  . For any two integers   n   and   k   with    1  ⩽  k  ⩽  n   , let[  n  ]    ⩽  kdenote the family of all subsets of    {  1  ,  ⋯  ,  n  }    that have at most   k   elements. We show that if   A   is a subfamily of[  m  ]    ⩽  r,   B   is a subfamily of[  n  ]    ⩽  s, and   A   and   B   are cross‐intersecting, then|    A    |    |    B    |   ⩽   ∑   i  =  1   r(m  −  1    i  −  1)∑   j  =  1   s(n  −  1    j  −  1),and equality holds if    A  =  {  A  ∈[  m  ]    ⩽  r:  1  ∈  A  }    and    B  =  {  B  ∈[  n  ]    ⩽  s:  1  ∈  B  }   .

A cross‐intersection theorem for subsets of a set