Prove that for each coloring of the points inside or on the boundary of a square with $1391$ colors, there exists a monochromatic regular hexagon.
Problem
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P1
Tags: combinatorics proposed, combinatorics
ocha
17.09.2012 05:40
Embed some sufficiently large triangular co-ordinate grid in the square, then employ Gallai's theorem where the subset in question are the vertices of a hexagon. (see here, page 12). Btw, what's the significance of 1391? It seems to appear in lots of Iranian problems.
goodar2006
17.09.2012 06:29
ocha wrote: Embed some sufficiently large triangular co-ordinate grid in the square, then employ Gallai's theorem where the subset in question are the vertices of a hexagon. (see here, page 12). Btw, what's the significance of 1391? It seems to appear in lots of Iranian problems. It's the current year in the Iranian Calender.
hamidkameli
07.07.2014 00:33
i can't understand the answer. could any one answer the question in detail.
SidVicious
30.05.2017 22:44
ocha wrote: Embed some sufficiently large triangular co-ordinate grid in the square, then employ Gallai's theorem where the subset in question are the vertices of a hexagon. (see here, page 12). When I open link, I see nothing (it says "not found"). Can anybody post the intended file about the theorem, or at least a solution?
dgrozev
31.05.2017 11:35
https://artofproblemsolving.com/community/c6h570463 The lattice $\mathbb{Z}^n$ can be replaced with any lattice.