Take the positive integers {1, 2, 3, ...} and color them red, green or blue. Is it true that no matter what coloring is chosen, we can always find three distinct numbers x, y and z so that x, y, z and x+y+z are all the same color?

This puzzle was inspired by: How do we find the numbers?

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    $\begingroup$ This puzzle implies the answer is yes, though the weaker condition here might allow a nicer and more accessible argument. $\endgroup$
    – xnor
    Commented May 22 at 8:26
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    $\begingroup$ With computer programs, I found that {1, ..., 93} allows a quad-free coloring but {1, ..., 94} doesn't. Doesn't feel like a useful insight though. $\endgroup$
    – Bubbler
    Commented May 22 at 9:07
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    $\begingroup$ @xnor I saw that puzzle after I posted my question and I debated whether I should delete my question or not. I decided to keep my question because I am hoping my weaker constraint can lead to an elegant solution. $\endgroup$ Commented May 22 at 9:38
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    $\begingroup$ @Moti The phrase, “no matter what coloring is chosen ...” is meant to include all possible colorings including all random colorings. $\endgroup$ Commented May 24 at 20:41
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    $\begingroup$ @Bubbler generalizing it into {1, ..., N} and K colours might be a better insight. $\endgroup$
    – Ver Nick
    Commented May 29 at 17:00


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