Number coloring

Probability Level 4

Is it possible to color every positive real number so that

• each number is colored one of 10 distinct colors, and
• any two numbers that differ in exactly one digit are colored differently?

For example, the ten numbers ${\color{maroon}0.111} \ldots,\, {\color{#D61F06}1.111}\ldots,\, {\color{#EC7300}2.111} \ldots,\, {\color{limegreen}3.111} \ldots,\, {\color{#20A900}4.111} \ldots,\, {\color{teal}5.111} \ldots,\, {\color{#3D99F6}6.111} \ldots,\, {\color{#0C6AC7}7.111} \ldots,\, {\color{#302B94}8.111} \ldots,\, {\color{magenta}9.111} \ldots$ must all be different colors because any two of them differ in exactly one digit. However, the numbers $10.000 \ldots$ and $1.000 \ldots$ do not have to be colored differently because they differ in more than one digit.

You may assume the axiom of choice holds.

×