Reverse mathematics of a color basis theorem

The infinite pigeonhole theorem asserts that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mi>m</mml:mi> </mml:math> is a function with a finite range, then there is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>j</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>m</mml:mi> </mml:math> such that the set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>∣</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>j</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:math> is infinite. This article uses the techniques of reverse mathematics and Weihrauch analysis to examine the strength of a theorem that finds all the values that occur infinitely often in the range of a function.