MathDB

Let

k

and

N

be positive real numbers which satisfy

k\leq N

. For

1\leq i \leq k

, there are subsets

A_i

\{1,2,3,\ldots,N\}

that satisfy the following property.
For arbitrary subset of

\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \}

A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}

is not an empty set.
Show that a subset

\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \}

exist that satisfies

n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k

. (

A \triangle B=A \cup B-A \cap B

)

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