Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Tagged graph-theory — 6 entries. Clear
Berge guessed perfection means no odd holes; forty years later Chudnovsky–Robertson–Seymour–Thomas proved it in ~150 pages. The capstone of perfection.
Jaeger's summit conjecture: every bridgeless cubic graph maps into the Petersen graph — implying cycle double cover, Berge–Fulkerson, and five-flow at once.
Colouring vertices and edges together: is maximum degree plus two always enough? Behzad asked in 1965; the exact bound is still open.
Does fractional edge-colouring pin down integral edge-colouring up to one colour? Goldberg and Seymour guessed yes in the 1970s; Chen–Jing–Zang proved it in 2019.
Fifty years open: can n nearly-disjoint n-sets always be coloured with just n colours? Proved for all large n in 2021 by Kang–Kelly–Kühn–Methuku–Osthus.
Four colours suffice for any map. The first major theorem proved by a computer — and the first mathematical controversy about what a proof is.
