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 0.111,1.111,2.111,3.111,4.111,5.111,6.111,7.111,8.111,9.111{\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 10.00010.000 \ldots and 1.0001.000 \ldots do not have to be colored differently because they differ in more than one digit.

You may assume the axiom of choice holds.


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