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?