One great problem, three kinds of immersion

From a story to your own conjecture

Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.

1 · DiscoverMeet the questionLearn when it appeared and why it still holds people’s attention.
2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
3 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.

Week 15 · ClassicalPartial progress

Hadwiger–Nelson — How Many Colors to Avoid Unit-Distance Monochromes?

Intermediate· Posed 1950

Problem

Color every point of the plane $\mathbb{R}^2$ so that any two points at distance exactly 1 get different colors. What is the minimum number of colors $\chi(\mathbb{R}^2)$ needed? Posed by Edward Nelson (1950).

Why it matters

The 7-vertex Moser spindle (1961) shows $\chi \ge 4$ , and a hexagonal tiling shows $\chi \le 7$ . For 60+ years the answer was known to be 4, 5, 6, or 7 — and only that.

Progress so far

On 2018-04-08 Aubrey de Grey proved $\chi\ge5$ with a 1581-vertex unit-distance graph. Polymath 16 reduced the witness to 553 vertices, and Jaan Parts later obtained a 509-vertex, 2442-edge graph. Minimizing a witness is distinct from determining the plane's exact chromatic number, which remains between 5 and 7.

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

Markdown + KaTeX supported (`$x^2$` inline, `$$\sum_{k=1}^n k$$` display)

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