TEST

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Test:

Alright, let’s put it in math form instead of a picture.


We’re talking about the unit distance graph of the plane, where:


<br>V(G) = \{ (x, y) \in \mathbb{R}^2 \}<br>

E(G) = { {u, v} \mid \text{distance}(u, v) = 1 }


The problem asks for:


<br>\chi(\mathbb{R}^2) = \min\{ k \in \mathbb{N} \mid \exists\ \text{proper } k\text{-coloring of } G \}<br>




Lower bound (Triangle/Moser idea):


Take a finite induced subgraph such that


<br>\chi(H) = 4<br>

That proves:


<br>\chi(\mathbb{R}^2) \ge 4<br>




Upper bound (Hexagon tiling):


Tile the plane with congruent hexagons of diameter .
Define


<br>f : \mathbb{R}^2 \to \{1,2,\dots,7\}<br>

<br>f(p) = (a(p) + 2b(p)) \bmod 7<br>

This guarantees no two points at distance exactly 1 have the same -value.
Therefore:


<br>\chi(\mathbb{R}^2) \le 7<br>




So mathematically:


<br>4 \le \chi(\mathbb{R}^2) \le 7<br>

“hexagon” = .





You basically jumped straight to:


<br>\boxed{\text{known bounds on the Hadwiger–Nelson problem}}<br>

If you want, I could also write the triangle forcing proof fully as inequalities — that’s the mathy equivalent of me “sketching” it. That’s where it gets spicy.

Edit: ChatGPT began to use Kanjis to explain "unsolvable maths problems using Infinity as a value" and was curious if said Kanjis would appear on here, but I guess not?
 
Last edited by JuanMena,
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