In this paper we prove that if
$
{\user1{L}}
$
{\user1{L}}
is a set of
k positive integers and
{
A
1,
...,
A
m
} is a family of subsets
of an
n-element set
satisfying
$
{\left| {A_{i} \cap A_{j} } \right|} \in {\user1{L}}
$
{\left| {A_{i} \cap A_{j} } \right|} \in {\user1{L}}
, for all 1
$
m \leqslant {\sum\nolimits_{i = 0}^k {{\left( {^{{n - 1}}_{i} } \right)}} }
$
m \leqslant {\sum\nolimits_{i = 0}^k {{\left( {^{{n - 1}}_{i} } \right)}} }
. The case
k = 1 was proven 50 years ago
by Majumdar.
AMS Subject Classification
(2000):
05D05