MEMO 2007 I2

A set of balls contains $n$ balls which are labeled with numbers $1,2,3,\ldots,n.$ We are given $k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that

(a) the balls labeled with the same number are of the same colour,

(b) any subset of $k+1$ balls with (not necessarily different) labels $a_{1},a_{2},\ldots,a_{k+1}$ satisfying the condition $a_{1}+a_{2}+\ldots+a_{k}= a_{k+1}$ , contains at least one ball of each colour.

Find, depending on $k$ the greatest possible number $n$ which admits such a colouring.