Color each positive integer with red and blue, whether there must be three numbers $a, b, c$ with same color such that $a^2+b^2 = c^2$? Such as if $3,4,5$ is blue, then $3^2 + 4^2 = 5^2$. Is there a color scheme so that there are no such three numbers?

wei zi
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    It is impossible to answer your question without a definition of "color". – poetasis Dec 13 '21 at 09:21
  • Sorry for my bad English, I mean every positive integer has a color, blue or red. This question was a question that came to mind after seeing Schur Theorem, but I couldn't figure it out after thinking about it for a long time. – wei zi Dec 13 '21 at 09:24
  • Thank you so much! I didn't expect it to be resolved so late. – wei zi Dec 13 '21 at 09:26
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    Perhaps there is a language translation problem. what do the colors red and blue mean in mathematics? – poetasis Dec 13 '21 at 09:36
  • @poetasis "Color" here is a standard term, and should be familiar to you from the four-color map theorem if nothing else. A "coloring" of a set is simply a function $c$ from that set to a set of "colors". We say that two elements $a$ and $b$ are colored the same if $c(a)=c(b)$. The elements of the set of "colors" have no further significance. – MJD Dec 13 '21 at 10:10
  • @weizi Your English is fine, and the question was perfectly clear. The other user is unfairly blaming you for their own lack of familiarity with this term. – MJD Dec 13 '21 at 10:15
  • @MJD I am not blaming anyone. It's just that I am a math amateur and I had no idea what color meant in math, even after reading about Boolean Pythagorean triples etc. I'm still not sure what they mean after your explanation but I will keep looking for other articles. – poetasis Dec 13 '21 at 12:00
  • If you don't understand something, it is more accurate, humble, and kind to start with a question like "What does 'color' mean here?" instead of a universal rejection like "It is impossible to answer your question". In this case the claim was mistaken, since it was _not_ impossible to answer the question. – MJD Dec 13 '21 at 12:47
  • @MJD The question was answerable because a definition existed. In my ignorance, I was asking for such a definition but I gave the wrong impression with my words. I’m dense sometimes and I’m still looking for an explanation that “I” can understand. Thanks for you input so far. – poetasis Dec 13 '21 at 16:25
  • What is a red number? What is a blue number? – poetasis Dec 20 '21 at 18:08

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