MathLabs

Problem 4

Let ABCABC be an equilateral triangle and let E\mathcal E be the union of its three sides. Determine whether every partition of E\mathcal E into two disjoint subsets has one subset containing the vertices of a right-angled triangle. Justify your answer.
Step 1 of 5: Projection lemma
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X,Y same color on one side⇒their orthogonal projections must have the opposite colorX,Y\text{ same color on one side}\Rightarrow\text{their orthogonal projections must have the opposite color}
Detailed analysis

Assume there is no monochromatic right triangle. If two points X,YX,Y on one side have the same color, then any point ZZ on another side whose orthogonal projection is one of these points cannot have that color whenever X,Y,ZX,Y,Z would form a right triangle. Otherwise those three points would be a forbidden monochromatic right triangle.