KöMaL 2021 A791

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: $i)$ For every combination of the switches the lighbulb emits a given color. $ii)$ If all switches are in a position with a given color, the lightbulb emits the same color. $iii)$ 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 $S$. ($U$ is an ultrafilter on $S$ if it satisfies the following: $a)$ The empty set is not in $U$. $b)$ If two sets are in $U$, their intersection is also in $U$. $c)$ If a set is in $U$, every subset of $S$ containing it are also in $U$. $d)$ Considering a set and its complement in $S$, exactly one of these sets is contained in $U$.) See also problem N.$\,$35. from the May issue of 1994 (in Hungarian). ( Problem and solution .)