MathDB

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\plus{}1 balls with (not necessarily different) labels a_{1},a_{2},\ldots,a_{k\plus{}1} satisfying the condition a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}, contains at least one ball of each colour. Find, depending on

k

the greatest possible number

n

which admits such a colouring.

Problem Statement