MathDB

A lightbulb is given that emits red, green or blue light and an infinite set

S

of switches, each with three positions labeled red, green and blue. We know the following:
[*]For every combination of the switches the lighbulb emits a given color. [*]If all switches are in a position with a given color, the lightbulb emits the same color. [*]If there are two combinations of the switches where each switch is in a different position, the lightbulb emits a different color for the two combinations.
We create the following set

U

containing some of the subsets of

S

: for each combination of the switches let us observe the color of the lightbulb, and put the set of those switches in

U

which are in the same position as the color of the lightbulb.
Prove that

U

is an ultrafilter on

S

. In other words, prove that

U

satisfies the following conditions:
[*]The empty set is not in

U.

[*]If two sets are in

U,

their intersection is also in

U.

[*]If a set is in

U,

every subset of

S

containing it is also in

U.

[*]Considering a set and its complement in

S,

exactly one of these sets is contained in

U.

Problem Statement