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A problem posed to me by a friend:

Show that any two-coloring of $\mathbb{R}^2$ that contains a monochromatic equilateral triangle of side-lengths 1 also contains monochromatic triangles of all side lengths $(1,a,b) \mid a+b>1$

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Presumably $a+b > 1$, otherwise there are no such triangles. –  Robert Israel May 15 '12 at 18:56
    
Some thoughts: Let the monochromatic equilateral triangle be $ABC$. There are lots of triangles you can make with side lengths $(1,a,b)$: you can use the sides $AB$, $AC$, or $BC$ as the length $1$ side and pick a corresponding point to get the other side lengths. With so many possibilities, perhaps you can show a monochromatic triangle can't be avoided. –  Austin Mohr May 15 '12 at 18:57
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This is posed as a difficult problem number 14.8 in Lovász László, Combinatorial Problems and Exercises, 2nd ed. You may check out the hints given there in the hints section. –  Zsbán Ambrus May 15 '12 at 19:10
    
Thanks for the reference, @ZsbánAmbrus –  Ternary May 15 '12 at 19:35
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