Equivalent subsets of a colored set

Let X be a finite set. Let φ be a function from X to the set of positive integers N . A pair ( X , φ ) is called a colored set. Two colored sets ( X 1 , φ 1 ) and ( X 2 , φ 2 ) are called equivalent if there exists a permutation σ of N such that | φ 1 − 1 ( y ) | = | φ 2 − 1 ( σ ( y ) ) | for any y ∈ N . We say that a colored set ( X , φ ) has a ( k ; l ) -partition if there exists a partition X = X 0 ∪ X 1 ∪ ⋯ ∪ X l such that | X i | = k for 1 ≤ i ≤ l , and ( X i , φ | X i ) and ( X j , φ | X j ) are equivalent for 1 ≤ i < j ≤ l . Let f ( k , l ) be the smallest integer n such that any colored set ( X , φ ) with | X | ≥ n has a ( k ; l ) -partition. It is shown that if k , l ≥ 2 with l ≥ k − 2 , then f ( k , l ) = ( k + 1 ) l − 1 .