Let
h
≥
2
h \geq 2
and
k
≥
1
k \geq 1
. It is proved that if
S
=
{
S
i
}
i
=
1
s
\mathcal {S} = \{ {S_i}\} _{i = 1}^s
and
T
=
{
T
j
}
j
=
1
t
\mathcal {T} = \{ {T_j}\} _{j = 1}^t
are two families of nonempty, pairwise disjoint sets such that
|
S
i
|
≤
h
,
|
T
j
|
≤
k
|{S_i}| \leq h,|{T_j}| \leq k
and
S
i
⊈
T
j
{S_i} \nsubseteq {T_j}
for all
i
i
and
j
j
, then the number
N
(
S
,
T
)
N(\mathcal {S},\mathcal {T})
of the sets
X
X
such that
X
X
is a minimal system of representatives for
S
\mathcal {S}
and
X
X
is simultaneously a system of representatives for
T
\mathcal {T}
that satisfies
N
(
S
,
T
)
≤
h
s
(
1
−
(
h
−
r
)
/
h
q
+
1
)
t
N(\mathcal {S},\mathcal {T}) \leq {h^s}{(1 - (h - r)/{h^{q + 1}})^t}
, where
k
=
q
(
h
−
1
)
+
r
k = q(h - 1) + r
with
0
≤
r
≤
h
−
2
0 \leq r \leq h - 2
. This was conjectured by M. B. Nathanson [3] in 1985.