Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.
Notable lists For over a century, various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions, with the Millennium Prize Problems each containing a reward of one million dollars.
Millennium Prize Problems Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:
Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP Riemann hypothesis Yang–Mills existence and mass gap The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003. However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.
Notebooks The Kourovka Notebook (Russian: Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since. The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since. The Dniester Notebook (Russian: Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory. The Erlagol Notebook (Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory.
Unsolved problems
Algebra
Birch–Tate conjecture on the relation between the order of the center of the Steinberg group of the ring of integers of a number field to the field's Dedekind zeta function. Casas-Alvero conjecture: if a polynomial of degree d {\displaystyle d} defined over a field K {\displaystyle K} of characteristic 0 {\displaystyle 0} has a factor in common with its first through d − 1 {\displaystyle d-1} -th derivative, then must f {\displaystyle f} be the d {\displaystyle d} -th power of a linear polynomial? Connes embedding problem in Von Neumann algebra theory Crouzeix's conjecture: the matrix norm of a complex function f {\displaystyle f} applied to a complex matrix A {\displaystyle A} is at most twice the supremum of | f ( z ) | {\displaystyle |f(z)|} over the field of values of A {\displaystyle A} . Determinantal conjecture on the determinant of the sum of two normal matrices. Eilenberg–Ganea conjecture: a group with cohomological dimension 2 also has a 2-dimensional Eilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} . Farrell–Jones conjecture on whether certain assembly maps are isomorphisms. Bost conjecture: a specific case of the Farrell–Jones conjecture Finite lattice representation problem: is every finite lattice isomorphic to the congruence lattice of some finite algebra? Goncharov conjecture on the cohomology of certain motivic complexes. Green's conjecture: the Clifford index of a non-hyperelliptic curve is determined by the extent to which it, as a canonical curve, has linear syzygies. Grothendieck–Katz p-curvature conjecture: a conjectured local–global principle for linear ordinary differential equations. Hadamard conjecture: for every positive integer k {\displaystyle k} , a Hadamard matrix of order 4 k {\displaystyle 4k} exists. Williamson conjecture: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices. Lander's conjecture: if an abelian group of order v {\displaystyle v} contains a difference set of order n {\displaystyle n} , and a prime p {\displaystyle p} divides both v {\displaystyle v} and n {\displaystyle n} , must the Sylow p {\displaystyle p} -subgroup be non-cyclic? This implies Ryser's cyclic difference-set conjecture, and hence the circulant Hadamard conjecture. Ryser's conjecture on circulant Hadamard matrices: no real circulant Hadamard matrix has order greater than 4. Barker sequence conjecture: no Barker sequence has length greater than 13; equivalently, none has even length greater than 4. Hadamard's maximal determinant problem: what is the largest determinant of a matrix with entries all equal to 1 or −1? Hilbert's fifteenth problem: put Schubert calculus on a rigorous foundation. Hilbert's sixteenth problem: what are the possible configurations of the connected components of M-curves? Homological conjectures in commutative algebra Jacobson's conjecture: the intersection of all powers of the Jacobson radical of a left-and-right Noetherian ring is precisely 0. Kaplansky's conjectures Köthe conjecture: if a ring has no nil ideal other than { 0 } {\displaystyle \{0\}} , then it has no nil one-sided ideal other than { 0 } {\displaystyle \{0\}} . Monomial conjecture on Noetherian local rings Existence of perfect cuboids and associated cuboid conjectures Pierce–Birkhoff conjecture: every piecewise-polynomial f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is the maximum of a finite set of minimums of finite collections of polynomials. Rota's basis conjecture: for matroids of rank n {\displaystyle n} with n {\displaystyle n} disjoint bases B i {\displaystyle B_{i}} , it is possible to create an n × n {\displaystyle n\times n} matrix whose rows are B i {\displaystyle B_{i}} and whose columns are also bases. Serre's conjecture II: if G {\displaystyle G} is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most 2 {\displaystyle 2} , then the Galois cohomology set H 1 ( F , G ) {\displaystyle H^{1}(F,G)} is zero. Serre's positivity conjecture that if R {\displaystyle R} is a commutative regular local ring, and P , Q {\displaystyle P,Q} are prime ideals of R {\displaystyle R} , then dim ( R / P ) + dim ( R / Q ) = dim ( R ) {\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)} implies χ ( R / P , R / Q ) > 0 {\displaystyle \chi (R/P,R/Q)>0} . Uniform boundedness conjecture for rational points: do algebraic curves of genus g ≥ 2 {\displaystyle g\geq 2} over number fields K {\displaystyle K} have at most some bounded number N ( K , g ) {\displaystyle N(K,g)} of K {\displaystyle K} -rational points? Wild problems: problems involving classification of pairs of n × n {\displaystyle n\times n} matrices under simultaneous conjugation. Zariski–Lipman conjecture: for a complex algebraic variety V {\displaystyle V} with coordinate ring R {\displaystyle R} , if the derivations of R {\displaystyle R} are a free module over R {\displaystyle R} , then V {\displaystyle V} is smooth. Zauner's conjecture: do SIC-POVMs exist in all dimensions?
Group theory
Andrews–Curtis conjecture: every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on relators and conjugations of relators Bounded Burnside problem: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite? Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems Herzog–Schönheim conjecture: if a finite system of left cosets of subgroups of a group G {\displaystyle G} form a partition of G {\displaystyle G} , then the finite indices of said subgroups cannot be distinct. The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals? Isomorphism problem of Coxeter groups Are there an infinite number of Leinster groups? Does generalized moonshine exist? Is every finitely presented periodic group finite? Is every group surjunctive? Problems in loop theory and quasigroup theory consider generalizations of groups
Representation theory Arthur's conjectures Dade's conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups. Demazure conjecture on representations of algebraic groups over the integers. Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras. McKay conjecture: in a group G {\displaystyle G} , the number of irreducible complex characters of degree not divisible by a prime number p {\displaystyle p} is equal to the number of irreducible complex characters of the normalizer of any Sylow p {\displaystyle p} -subgroup within G {\displaystyle G} .
Analysis
The Brennan conjecture: estimating the integral powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of C {\displaystyle \mathbb {C} }
Fuglede's conjecture on whether nonconvex sets in R {\displaystyle \mathbb {R} } and R 2 {\displaystyle \mathbb {R} ^{2}} are spectral if and only if they tile by translation. Goodman's conjecture on the coefficients of multivalued functions Invariant subspace problem – does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself? Kung–Traub conjecture on the optimal order of a multipoint iteration without memory Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials The mean value problem: given a complex polynomial f {\displaystyle f} of degree d ≥ 2 {\displaystyle d\geq 2} and a complex number z {\displaystyle z} , is there a critical point c {\displaystyle c} of f {\displaystyle f} such that | f ( z ) − f ( c ) | ≤ | f ′ ( z ) | | z − c | {\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|} ? The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy Vitushkin's conjecture on compact subsets of C {\displaystyle \mathbb {C} } with analytic capacity 0 {\displaystyle 0}
What is the exact value of Landau's constants, including Bloch's constant? Regularity of solutions of Euler equations Convergence of Flint Hills series Regularity of solutions of Vlasov–Maxwell equations Are there infinitely many Lehmer pairs?
Combinatorics
The 1/3–2/3 conjecture – does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3? The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition Problems in Latin squares – open questions concerning Latin squares The lonely runner conjecture – if k {\displaystyle k} runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1 / k {\displaystyle 1/k} from each other runner) at some time? Superpermutations - smallest possible string of n digits that contains all possible permutations of n Map folding – various problems in map folding and stamp folding. No-three-in-line problem – how many points can be placed in the n × n {\displaystyle n\times n} grid so that no three of them lie on a line? Rudin's conjecture on the number of squares in finite arithmetic progressions The sunflower conjecture – can the number of k {\displaystyle k} size sets required for the existence of a sunflower of r {\displaystyle r} sets be bounded by an exponential function in k {\displaystyle k} for every fixed r > 2 {\displaystyle r>2} ? Frankl's union-closed sets conjecture – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets Give a combinatorial interpretation of the Kronecker coefficients The size m ( n ) {\displaystyle m(n)} of the smallest collection of n {\displaystyle n} -uniform sets without Property B for n ≥ 5 {\displaystyle n\geq 5}
The values of the Dedekind numbers M ( n ) {\displaystyle M(n)} for n ≥ 10 {\displaystyle n\geq 10}
The values of the Ramsey numbers, particularly R ( 5 , 5 ) {\displaystyle R(5,5)}
The values of the Van der Waerden numbers Finding a function to model n-step self-avoiding walks
Dynamical systems
Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory. Berry–Tabor conjecture in quantum chaos Banach's problem – is there an ergodic system with simple Lebesgue spectrum? Birkhoff conjecture – if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse? Collatz conjecture (also known as the 3 n + 1 {\displaystyle 3n+1} conjecture) Eden's conjecture that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an unstable periodic orbit embedded into the attractor. Fatou conjecture that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. Furstenberg conjecture – is every invariant and ergodic measure for the × 2 , × 3 {\displaystyle \times 2,\times 3} action on the circle either Lebesgue or atomic? Kaplan–Yorke conjecture on the dimension of an attractor in terms of its Lyapunov exponents Margulis conjecture – measure classification for diagonalizable actions in higher-rank groups. Hilbert–Arnold problem – is there a uniform bound on limit cycles in generic finite-parameter families of vector fields on a sphere? MLC conjecture – is the Mandelbrot set locally connected? Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits. Quantum unique ergodicity conjecture on the distribution of large-frequency eigenfunctions of the Laplacian on a negatively-curved manifold Rokhlin's multiple mixing problem – are all strongly mixing systems also strongly 3-mixing? Triangular billiards – does every triangle have a periodic billiards path? Weinstein conjecture – does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow? Does every positive integer generate a juggler sequence terminating at 1? Lyapunov function: Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion? Is every reversible cellular automaton in three or more dimensions locally reversible?
Games and puzzles
Combinatorial games
Sudoku: How many puzzles have exactly one solution? How many puzzles with exactly one solution are minimal? What is the maximum number of givens for a minimal puzzle? Tic-tac-toe variants: Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also Hales–Jewett theorem and nd game) Chess: What is the outcome of a perfectly played game of chess? (See also first-move advantage in chess) Go: What is the perfect value of Komi? Set: What is the largest possible cap set, as a function of n {\displaystyle n} in the n {\displaystyle n} -dimensional affine space over the three-element field? Are the nim-sequences of all finite octal games eventually periodic? Is the nim-sequence of Grundy's game eventually periodic?
Games with imperfect information Rendezvous problem
Geometry
Algebraic geometry
Abundance conjecture: if the canonical bundle of a projective variety with Kawamata log terminal singularities is nef, then it is semiample. Bass conjecture on the finite generation of certain algebraic K-groups. Bass–Quillen conjecture relating vector bundles over a regular Noetherian ring and over the polynomial ring A [ t 1 , … , t n ] {\displaystyle A[t_{1},\ldots ,t_{n}]} . Deligne conjecture: any one of numerous named for Pierre Deligne. Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex. Dixmier conjecture: any endomorphism of the Weyl algebras A 1 {\displaystyle A_{1}} and A 2 {\displaystyle A_{2}} is an automorphism. Fröberg conjecture on the Hilbert functions of a set of forms. Fujita conjecture regarding the line bundle K M ⊗ L ⊗ m {\displaystyle K_{M}\otimes L^{\otimes m}} constructed from a positive holomorphic line bundle L {\displaystyle L} on a compact complex manifold M {\displaystyle M} and the canonical line bundle K M {\displaystyle K_{M}} of M {\displaystyle M}
General elephant problem: do general elephants have at most Du Val singularities? Hartshorne's conjectures In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent? Jacobian conjecture for two-dimensional spaces: if a two-dimensional polynomial mapping over a characteristic-0 field has a constant nonzero Jacobian determinant, then does it have a regular (i.e. with polynomial components) inverse function? Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities. Nagata–Biran conjecture that if X {\displaystyle X} is a smooth algebraic surface and L {\displaystyle L} is an ample line bundle on X {\displaystyle X} of degree d {\displaystyle d} , then for sufficiently large r {\displaystyle r} , the Seshadri constant satisfies ε ( p 1 , … , p r ; X , L ) = d / r {\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}} . Nakai conjecture: if a complex algebraic variety has a ring of differential operators generated by its contained derivations, then it must be smooth. Parshin's conjecture: the higher algebraic K-groups of any smooth projective variety defined over a finite field must vanish up to torsion. Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields k {\displaystyle k} to the Galois group of k {\displaystyle k} . Standard conjectures on algebraic cycles Tate conjecture on the connection between algebraic cycles on algebraic varieties and Galois representations on étale cohomology groups. Virasoro conjecture: a certain generating function encoding the Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points Are infinite sequences of flips possible in dimensions greater than 3? Resolution of singularities in characteristic p {\displaystyle p}
Covering and packing Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set. The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be? The Erdős–Oler conjecture: when n {\displaystyle n} is a triangular number, packing n − 1 {\displaystyle n-1} circles in an equilateral triangle requires a triangle of the same size as packing n {\displaystyle n} circles. The disk covering problem about finding the smallest real number r ( n ) {\displaystyle r(n)} such that n {\displaystyle n} disks of radius r ( n ) {\displaystyle r(n)} can be arranged in such a way as to cover the unit disk. The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24 Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions. Square packing in a square: what is the asymptotic growth rate of wasted space? Ulam's packing conjecture about the identity of the worst-packing convex solid The Tammes problem for numbers of nodes greater than 14 (except 24).
Differential geometry
The spherical Bernstein's problem, a generalization of Bernstein's problem Carathéodory conjecture: any convex, closed, and twice-differentiable surface in three-dimensional Euclidean space admits at least two umbilical points. Cartan–Hadamard conjecture: can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds? Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes. Chern's conjecture for hypersurfaces in spheres, a number of closely related conjectures. Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed. The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds Osserman conjecture: that every Osserman manifold is either flat or locally isometric to a rank-one symmetric space Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of S n + 1 {\displaystyle S^{n+1}} is n {\displaystyle n} .
Discrete geometry
The big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point sets The Dirac–Motzkin conjecture on the minimum number of ordinary lines for n points in the Euclidean plane The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies Solving the happy ending problem for arbitrary n {\displaystyle n}
Improving lower and upper bounds for the Heilbronn triangle problem. Kalai's 3d conjecture on the least possible number of faces of centrally symmetric polytopes. The Kobon triangle problem on triangles in line arrangements The Kusner conjecture: at most 2 d {\displaystyle 2d} points can be equidistant in L 1 {\displaystyle L^{1}} spaces The McMullen problem on projectively transforming sets of points into convex position Opaque forest problem on finding opaque sets for various planar shapes Orchard-planting problem on the maximum number of 3-point lines attainable by a configuration of n points in the plane How many unit distances can be determined by a set of n points in the Euclidean plane? Finding matching upper and lower bounds for k-sets and halving lines For each arrangement of points in which the rectilinear crossing number is minimized, is the number of halving lines maximized? Tripod packing: how many tripods can have their apexes packed into a given cube?
Euclidean geometry
The Atiyah conjecture on configurations on the invertibility of a certain n {\displaystyle n} -by- n {\displaystyle n} matrix depending on n {\displaystyle n} points in R 3 {\displaystyle \mathbb {R} ^{3}}
Bellman's lost-in-a-forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link? Connelly’s blooming conjecture: Does every net of a convex polyhedron have a blooming? Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist? Dissection into orthoschemes – is it possible for simplices of every dimension? Ehrhart's volume conjecture: a convex body K {\displaystyle K} in n {\displaystyle n} dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than ( n + 1 ) n / n ! {\displaystyle (n+1)^{n}/n!}
Falconer's conjecture: sets of Hausdorff dimension greater than d / 2 {\displaystyle d/2} in R d {\displaystyle \mathbb {R} ^{d}} must have a distance set of nonzero Lebesgue measure The values of the Hermite constants for dimensions other than 1–8 and 24 What is the lowest number of faces possible for a holyhedron? Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem – does every Jordan curve have an inscribed square? The Kakeya conjecture – do n {\displaystyle n} -dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to n {\displaystyle n} ? The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar. In Meissner body): Are the two Meissner tetrahedra the minimum-volume three-dimensional shapes of constant width? Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane? The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor? In parallelohedron: Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as a parallelohedron? Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram? Ropelength problems: Is there a general expression for the minimum ropelength of an arbitrary closed knot? What constant 1.1 < a ≤ 10.76 {\displaystyle 1.1<a\leq 10.76} governs the lower bound of a closed knot K {\displaystyle K} 's minimum ropelength L ( K ) ≥ a Cr ( K ) 3 / 4 {\displaystyle L(K)\geq a\operatorname {Cr} (K)^{3/4}} ? Is the upper bound of a closed knot's minimum ropelength linear to its crossing number? Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it? Does every convex polyhedron have Rupert's property? Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding? Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other? The Thomson problem – what is the minimum energy configuration of n {\displaystyle n} mutually-repelling particles on a unit sphere? Convex uniform 5-polytopes – find and classify the complete set of these shapes
Non-Euclidean geometry
Hilbert's third problem for non-Euclidean geometries: in spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?
Graph theory
Algebraic graph theory Babai's problem: which groups are Babai invariant groups? Brouwer's conjecture on upper bounds for sums of eigenvalues of Laplacians of graphs in terms of their number of edges
Games on graphs γ–θ conjecture: Does there exist a graph G {\displaystyle G} such that the dominating number, eternal domination number, and clique covering number satisfy γ ( G ) = γ ∞ ( G ) < θ ( G ) {\displaystyle \gamma (G)=\gamma _{\infty }(G)<\theta (G)} ? Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs Meyniel's conjecture that cop number is O ( n ) {\displaystyle O({\sqrt {n}})}
Suppose Alice has a winning strategy for the vertex coloring game on a graph G {\displaystyle G} with k {\displaystyle k} colors. Does she have one for k + 1 {\displaystyle k+1} colors?
Graph coloring and labeling
The 1-factorization conjecture that if n {\displaystyle n} is odd or even and k ≥ n , n − 1 {\displaystyle k\geq n,n-1} , respectively, then a k {\displaystyle k} -regular graph with 2 n {\displaystyle 2n} vertices is 1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization. Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs The Earth–Moon problem: what is the maximum chromatic number of biplanar graphs? The Erdős–Faber–Lovász conjecture on coloring unions of cliques The graceful tree conjecture that every tree admits a graceful labeling Rosa's conjecture that all triangular cacti are graceful or nearly-graceful The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree The Hadwiger conjecture relating coloring to clique minors The Hadwiger–Nelson problem on the chromatic number of unit distance graphs Jaeger's Petersen-coloring conjecture: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph The list coloring conjecture: for every graph, the list chromatic index equals the chromatic index The overfull conjecture that a graph with maximum degree Δ ( G ) ≥ n / 3 {\displaystyle \Delta (G)\geq n/3} is class 2 if and only if it has an overfull subgraph S {\displaystyle S} satisfying Δ ( S ) = Δ ( G ) {\displaystyle \Delta (S)=\Delta (G)} . The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree
Graph drawing and embedding The Albertson conjecture: the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number Conway's thrackle conjecture that thrackles cannot have more edges than vertices The GNRS conjecture on whether minor-closed graph families have ℓ