Colouring vertices and edges together: is maximum degree plus two always enough? Behzad asked in 1965; the exact bound is still open.
The Problems
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.
Difficulty 4/5 — 50 shown. Clear
The Hanna Neumann conjecture
historic
How large can the intersection of two free subgroups be? Hanna Neumann asked in 1957; Mineyev and Friedman proved the sharp bound independently in 2011.
The Ramanujan–Nagell equation
historic
Ramanujan guessed in 1913 that x² + 7 = 2ⁿ has exactly five solutions; Nagell proved him right in 1948 via the arithmetic of Q(√−7).
Are 8 and 9 the only consecutive perfect powers? Catalan asked in 1844; Mihăilescu proved it in 2002 with a short cyclotomic argument.
The Erdős–Faber–Lovász conjecture
historic
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.
Are all hyperbolic groups residually finite? Gromov's basic question — no counterexample known, no proof in sight.
Short vectors balance to constant discrepancy with the right signs. Conjectured O(1), proved O(√log n) — the central open balancing statement.
Pillai's Conjecture
open
Gaps between perfect powers grow without bound. Catalan settled gap 1 (8 and 9); every other fixed gap is open, and abc would imply them all.
Subcomplexes of aspherical 2-complexes stay aspherical. Open since 1941; LOT complexes from ribbon discs are the test cases.
Suslin's Problem
open
Does the countable chain condition characterise ℝ? Independent of ZFC: Suslin lines exist under V = L, consistently none exist (Solovay–Tennenbaum 1971).
All NP-complete problems are the same problem up to polynomial-time recoding. Open since 1977; a positive answer would reveal deep structure in NP.
How evenly can s points spread before some triangle gets tiny? Exact growth of the optimal minimal-triangle area is open between log s/s² and n^-7/6.
The MDS Conjecture
open
How long can an optimal error-correcting code be? The MDS conjecture caps q-ary codes at length q+1 — proved for prime alphabets, open in general.
Which integers are areas of rational right triangles? Tunnell's criterion decides it assuming BSD; unconditionally, no general method is known.
Exact van der Waerden numbers: how long an interval forces a monochromatic k-term progression? Only scattered values known; W(2,7) already open.
A polynomial sharing a factor with every derivative must be a power of a linear one. Open since 2001; settled for prime-power degrees, open in general.
Below √n, no efficient algorithm finds a planted clique in a random graph. The hardness assumption behind sparse PCA, community detection and average-case crypto.
Does BPP = P?
open
Is randomness essential to efficient computation? BPP = P would follow from strong circuit lower bounds; unconditionally, the question is wide open.
Randomisation achieves polylog(k)-competitiveness for k-server on every metric, independent of the space size. Best known bounds still depend on n; the k-only bound is open.
Borel's Conjecture
open
Every strong measure zero set of reals is countable. Independent of ZFC like the continuum hypothesis: consistent (Laver 1976), refuted under CH (Sierpiński 1928).
Can the product of two 2-spheres carry everywhere-positive curvature? Hopf's 1930s question — no example found, no obstruction proved.
The Tammes Problem
open
How evenly can N points spread on a sphere? The botanist's problem behind spherical codes — exact optima known only for scattered N, including the kissing twelve.
How many equal spheres can kiss a central one? Solved in dimensions 1, 2, 3, 4, 8 and 24 only — every other dimension, starting with 5, remains open.
Each half of the interval between consecutive squares contains a prime. Stronger than Legendre's conjecture, from which it directly implies the one-prime-per-square case.
Andrica's Conjecture
open
The square roots of consecutive primes always differ by less than one. A prime-gap conjecture strictly stronger than Legendre's, open since 1986.
There is always a prime between consecutive perfect squares. The oldest open problem about prime gaps — weaker than Oppermann's, Andrica's and Cramér's conjectures, and implied by each of them.
Hadwiger–Nelson covers ℝ² (between 5 and 7). What about ℝ³, ℝ⁴, and beyond? Even the growth RATE of χ(ℝ^d) as d increases is unknown within exponential factors.
Lee spheres tile ℤ^n perfectly for n ≤ 2 and diameter-specific cases. Whether perfect Lee codes exist in higher dimensions beyond known families is one of coding theory's oldest open questions.
Does a projective plane of order 10 exist? No (proven by computation). Order 12? No. But whether planes exist for ALL non-prime-power orders satisfying Bruck–Ryser remains open — order 12 is the smallest unresolved case after the n=10 computation.
How many self-avoiding walks of length n exist on a lattice? The growth rate (connective constant) is unknown even for the square lattice — Duminil-Copin and Smirnov solved the hexagonal case exactly in 2010, but every other lattice resists.
Gauss's Class Number One Problem
historic
Exactly nine imaginary quadratic fields have class number one. Gauss listed them in 1801; proving his list complete took 166 years, involved a solution ignored for twenty years, and required computational verification beyond human capability.
The Green–Tao Theorem
historic
The primes contain arithmetic progressions of EVERY finite length. Euler noticed prime patterns in 1770; Green and Tao proved arbitrarily long ones exist — combining Szemerédi's theorem with a transference principle.
Can every balanced presentation of the trivial group be reduced to the trivial presentation by Nielsen moves plus conjugations? Connected to the smooth 4-dimensional Poincaré conjecture via handlebody calculus.
Vaught's Conjecture
open
A complete theory's countable models are either countably infinite in number or exactly continuum-many — never something in between. Model theory's deepest unresolved classification question.
Every ribbon knot bounds a singular disc with only self-intersections of one type. Does every SLICE knot (bounding a disc in 4-space) also bound such a ribbon? The first test case for distinguishing smooth from topological 4D knot theory.
Bond percolation on the square lattice has critical probability exactly 1/2. Harris proved no percolation below; Kesten closed the gap twenty-three years later — founding rigorous percolation theory.
Every high-dimensional normed space contains a subspace of dimension → ∞ that is ALMOST Euclidean. The theorem that launched asymptotic geometric analysis.
Are infinitely many supersingular elliptic curves over ℚ? Kaneko–Zagier conjectured a precise count formula; the answer controls deep connections between modular forms and crystallographic groups.
The Generalised Continuum Hypothesis asks whether \(2^{ℵ_α}\) = ℵ_{α+1} at EVERY level of the cardinal hierarchy. GCH implies CH, so it inherits the same independence — but large cardinals may change the story at higher levels.
Make inscribed-square's affine cousin: does every centrally symmetric convex body contain an inscribed affine-regular hexagon? A test case for understanding symmetric structures inside asymmetric containers.
If a family of convex sets has the property that among any p sets, some q intersect, how few points pierce the entire family? The (3,2) case is the classical (p,q)-theorem; the tight bound is open for most parameters.
The Collatz map extended to negative integers produces additional cycles beyond 0 and -1. Classifying ALL cycles of the generalised 3x±1 map is a harder cousin of the original problem.
Lehmer's Conjecture
open
Can integer polynomials have Mahler measure arbitrarily close to 1 without equalling it? Lehmer's degree-10 polynomial holds the world record ≈ 1.17628 — unbeaten since 1933.
Which closed subsets of [0,1] are invariant under both doubling and tripling mod 1? Only the trivial ones should exist — fifty-plus years of partial rigidity and the general case stands.
The Mertens Conjecture
historic
|M(n)| < √n for all n, where M is the Mertens function? A conjecture implying RH — disproved in 1985 by computation so indirect that the first counterexample remains beyond reach even now.
Hilbert's Fifth Problem
historic
Are continuous transformation groups automatically differentiable — i.e., is every locally Euclidean topological group a Lie group? Yes: solved in 1952, with a beautiful twist left open in the non-Archimedean world.
Whitehead's Problem
historic
Is every abelian group A with Ext(A, ℤ) = 0 free? Shelah's answer: YES and NO — provably independent of the standard axioms. Set-theoretic pluralism's second monument after CH.
Does the prime-counting function ever outrun the logarithmic integral? Gauss's tables said never. Littlewood proved the lead changes hands infinitely often — and nobody knows where the FIRST flip happens beyond astronomical bounds.
Serve requests arriving online with k mobile servers at minimum movement cost. Can any deterministic algorithm achieve k-competitiveness against the optimal offline server placement on EVERY metric space?
Danzer's Problem
open
Is there a point set of bounded density that intersects every convex body of volume 1? Sixty years of constructions either hit everything too sparsely or grow exponentially.