Color every point of an infinite sheet so that no two points exactly one unit apart are the same color. What is the fewest colors that works? The answer is 5, 6, or 7 — and no one on Earth knows which.
Here is one of the most embarrassing open problems in mathematics — embarrassing because a child can understand it and no one can solve it. Take the whole infinite plane. Color every single point. The only rule: any two points exactly distance 1 apart must get different colors. How few colors do you need?
It sounds like it should have a tidy answer. It was first asked by Edward Nelson in 1950, when he was eighteen. Within that same year the answer was pinned between 4 and 7. And there it sat. For sixty-eight years nobody could move either number — until 2018, when the lower bound finally ticked up by one, to 5, in a paper by a biologist better known for wanting to cure aging. The gap is still open. The number you are looking for is 5, 6, or 7, and which of the three is one of the genuinely unsolved questions of geometry.
This page is an instrument for both ends of that gap. First the lower bound — why you need at least a certain number — which you can feel by losing to it. Then the upper bound — why seven always suffice — which you can watch work. And in between, the strange 2018 jump, and the stranger fact that the answer might depend on the axioms of set theory.
How could anyone prove you need more than three colors, when there are infinitely many points? You don't have to handle all of them. You only need to find a finite set of points whose required distances trap you. The most famous such trap is seven points called the Moser spindle (the Moser brothers, 1961). Every edge drawn below joins two points that are exactly distance 1 apart — so the two ends must differ in color. Try to color the seven points with three colors so that no edge has both ends the same. Tap a color, then tap a point.
You cannot do it with three — and that is not an opinion. Press check all colorings and the page enumerates every one of the 3⁷ = 2,187 ways to paint these seven points with three colors and reports how many obey the distance rule. The count is zero. Switch to four colors and suddenly it is easy. Seven points, eleven unit distances, and three colors are provably not enough: therefore the whole plane needs at least four, because the plane contains this little constellation wherever you care to place it. (The spindle is exactly four-critical: remove any single edge and three colors return — it is a minimal trap, nothing wasted.)
Why is seven always enough? Because someone exhibited an actual coloring of the entire plane that uses seven colors and never breaks the rule — found by John Isbell, also in 1950. Tile the plane with hexagons a little smaller than one unit across, and color them in a repeating seven-color pattern. Any two points inside the same tile are closer than a unit, so they're safe; and the pattern is arranged so that two tiles of the same color are always more than a unit apart, so points one unit away always land on a different color. Move your cursor (or finger) over the tiling: the ring shows every point exactly distance 1 away. It never touches your color.
That is the whole classical story, and it is a strange one: the upper bound, 7, has not moved since 1950. Every advance for seventy-five years has been at the bottom, clawing the lower bound upward one integer at a time.
For sixty-eight years the lower bound stuck at four — the spindle's verdict. Then in April 2018 Aubrey de Grey — a biologist, an amateur in this field — posted a paper titled, with no flourish, “The chromatic number of the plane is at least 5.” He had built a single finite unit-distance graph, like the spindle but vastly larger, that cannot be colored with four colors. His graph had 1,581 points. Because it lives in the plane, the plane inherits its verdict: at least five colors.
No human checked 1,581 points by hand. The graph's 4-uncolorability was confirmed by SAT solvers — the same automated reasoning engines used to verify chips — and independently reproduced by other researchers with different software. Then the internet went to work shrinking it: a Polymath collaboration and the SAT specialist Marijn Heule drove the example down and down, and in 2020 Jaan Parts reached a 5-chromatic graph of just 509 points — the smallest known. From the spindle's 7 to de Grey's 1,581 to Parts's 509: the trap got bigger to catch a fifth color, then was filed down toward the bone.
The upper bound has not moved since 1950. We have spent seventy-five years proving the answer is bigger than four — and we still cannot rule out seven.
There is one more reason this problem is hard, and it is the strangest of all: the answer may depend on which axioms of mathematics you accept. The proof that the plane's chromatic number is even attained on a finite graph (the fact the whole spindle strategy relies on) uses the Axiom of Choice. And in 2003, Saharon Shelah and Alexander Soifer exhibited a related distance graph — on the line, not the plane — whose chromatic number is 2 if you accept Choice but becomes uncountably infinite in a universe where every set of reals is measurable. For the plane itself the dependence is not a settled theorem, but the warning stands: this may be a question whose answer is not a fact about geometry so much as a fact about which mathematics you are standing in. (Earlier, in 1981, Falconer showed that if you insist the color regions be measurable, the plane needs at least five — a hint of the 2018 result, decades early, under an extra rule.)
So the honest state of the art: 5 ≤ χ ≤ 7. The plane needs at least five colors and at most seven. Three numbers; one of them is right; the proof of which is not merely missing but possibly entangled with the foundations of mathematics. You can hold the entire problem in your head in a minute, and you can play with both of its walls on this page — and the room between them has stood empty for seventy-five years.