# Can we find four identically colored numbers?

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?

• This puzzle implies the answer is yes, though the weaker condition here might allow a nicer and more accessible argument.
– xnor
Commented May 22 at 8:26
• 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. Commented May 22 at 9:07
• @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. Commented May 22 at 9:38
• @Moti The phrase, “no matter what coloring is chosen ...” is meant to include all possible colorings including all random colorings. Commented May 24 at 20:41
• @Bubbler generalizing it into {1, ..., N} and K colours might be a better insight. Commented May 29 at 17:00