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 5 of 5: Contradiction and conclusion
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three pairwise different colors are impossible with two colors\text{three pairwise different colors are impossible with two colors}
Detailed analysis

There are only two subsets (colors), so three points cannot be pairwise differently colored. This contradiction proves that every two-partition of E\mathcal E contains a monochromatic right-angled triangle.