The seventh problem, the Poincaré conjecture, has been solved. The smooth four-dimensional Poincaré conjecture is still unsolved. That is, can a four-dimensional topological sphere have two or more inequivalent smooth structures?
Finding a formula for the probability that two elements chosen at random generate the symmetric group
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
The Lonely runner conjecture: if runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance from each other runner) at some time?
Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
Deriving a closed-form expression for the percolation threshold values, especially (square site)
Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow
Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality does it have a model of cardinality continuum?^{[19]}
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^{[20]}
If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?^{[21]}^{[22]}
Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
Kueker's conjecture^{[23]}
Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
Lachlan's decision problem
Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?^{[24]}
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?^{[25]}
The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
Assorted sphere packing problems, e.g. the densest irregular hypersphere packings
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.^{[28]}
^Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338.
Fan Chung; Ron Graham (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN1-56881-111-X.
Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN0-387-97506-3.
Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer. ISBN0-387-20860-7.
Victor Klee; Stan Wagon (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN0-88385-315-9.
Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN0-06-093558-8.
John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN0-309-08549-7.
Keith Devlin (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN978-0-7607-8659-8.
Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN0-691-11748-9.