The Problem Mathematicians Abandoned — Until an Outsider Wouldn't Let It Go

How many colors are needed to color every point of an infinite plane if points exactly one unit apart can never match? For decades the Hadwiger–Nelson problem was trapped between four and seven. Then in 2018 biogerontologist Aubrey de Grey combined copies of the familiar Moser spindle into a 1,581-vertex unit-distance graph that could not be colored with four colors. The breakthrough raised the lower bound to five and triggered an open collaborative search that eventually reduced the construction to 509 vertices.

The Hadwiger–Nelson problem turns a rule simple enough for a child into one of graph theory's most stubborn open questions. Its history shows how finite graphs can constrain the coloring of an infinite plane, why the seven-vertex Moser spindle mattered for decades, and how Aubrey de Grey unexpectedly broke a 68-year stalemate in 2018. The story then follows SAT solvers, Polymath collaboration and Jaan Parts's 509-vertex graph—while the final answer stubbornly remains one of only three possibilities: five, six or seven.