Suppose the
r
r
-subsets of an
n
n
-element set are colored by
t
t
colors. THEOREM 1.1. If
n
≥
(
t
−
1
)
(
k
−
1
)
+
k
⋅
r
n \geq (t - 1)(k - 1) + k \cdot r
, then there are
k
k
pairwise disjoint
r
r
-sets having the same color. This was conjectured by Erdös
[
E
]
[{\mathbf {E}}]
in 1973. Let
T
(
n
,
r
,
s
)
T(n,\,r,\,s)
denote the Turán number for
s
s
-uniform hypergraphs (see
§
1
\S 1
). THEOREM 1.3. If
ε
>
0
,
t
≤
(
1
−
ε
)
T
(
n
,
r
,
s
)
/
(
k
−
1
)
\varepsilon > 0,\,t \leq (1 - \varepsilon )T(n,\,r,\,s)/(k - 1)
, and
n
>
n
0
(
ε
,
r
,
s
,
k
)
n > {n_0}(\varepsilon ,\,r,\,s,\,k)
, then there are
k
k
r
r
-sets
A
1
,
A
2
,
…
,
A
k
{A_1},{A_2}, \ldots ,{A_k}
having the same color such that
|
A
i
∩
A
j
|
>
s
\left | {{A_i} \cap {A_j}} \right | > s
for all
1
≤
i
>
j
≤
k
1 \leq i > j \leq k
. If
s
=
2
,
ε
s = 2,\,\varepsilon
can be omitted. Theorem 1.1 is best possible. Its proof generalizes Lovász’ topological proof of the Kneser conjecture (which is the case
k
=
2
k = 2
). The proof uses a generalization, due to Bárány, Shlosman, and Szücs of the Borsuk-Ulam theorem. Theorem 1.3 is best possible up to the
ε
\varepsilon
-term (for large
n
n
). Its proof is purely combinatorial, and employs results on kernels of sunflowers.