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.