Question 118
A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in each cell) of a $k \times n$ table ($k$ is the number of rows of the table, $n$ is the number of its columns). We call a table *nice* if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R_1$ and $R_2$ there exists a nonempty set $S$ of columns such that on the intersection of any row from $R_1$ with the columns from $S$ there is an even number of $1$'s, while on the intersection of any row from $R_2$ with the columns from $S$ there is an odd number of $1$'s.
Find the greatest number $k$ such that there exists at least one nice $k \times n$ table.
Find the greatest number $k$ such that there exists at least one nice $k \times n$ table.