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Coloring points of a square, finding a monochromatic hexagon

Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P1

September 15, 2012
combinatorics proposedcombinatorics

Problem Statement

Prove that for each coloring of the points inside or on the boundary of a square with 13911391 colors, there exists a monochromatic regular hexagon.