# List of unsolved problems in mathematics

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved.[1] These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.

## Lists of unsolved problems in mathematics

Over the course of time, several lists of unsolved mathematical problems have appeared.

ListNumber of problemsNumber unresolved
or incompletely resolved
Proposed byProposed in
Hilbert's problems[2]2315David Hilbert1900
Landau's problems[3]44Edmund Landau1912
Taniyama's problems[4]36-Yutaka Taniyama1955
Thurston's 24 questions[5][6]24-William Thurston1982
Smale's problems1814Stephen Smale1998
Millennium Prize problems76[7]Clay Mathematics Institute2000
Simon problems15<12[8][9]Barry Simon2000
Unsolved Problems on Mathematics for the 21st Century[10]22-Jair Minoro Abe, Shotaro Tanaka2001
DARPA's math challenges[11][12]23-DARPA2007

### Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of 2019:[7]

The seventh problem, the Poincaré conjecture, has been solved.[13] The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[14]

## Unsolved problems

### Graph theory

#### Word-representation of graphs

• Characterise (non-)word-representable planar graphs
• Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs [92])
• Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter
• Is the line graph of a non-word-representable graph always non-word-representable?
• Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?
• Is it true that out of all bipartite graphs crown graphs require longest word-representants?
• Characterise word-representable graphs in terms of (induced) forbidden subgraphs.
• Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?

### Model theory and formal languages

• Vaught's conjecture
• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ${\displaystyle \aleph _{0}}$ is a simple algebraic group over an algebraically closed field.
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for ${\displaystyle \aleph _{1}}$-saturated models of a countable theory.[109]
• Determine the structure of Keisler's order[110][111]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p}}$ decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$?
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[112]
• The Stable Forking Conjecture for simple theories[113]
• For which number fields does Hilbert's tenth problem hold?
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ${\displaystyle \aleph _{\omega _{1}}}$ does it have a model of cardinality continuum?[114]
• Shelah's eventual categoricity conjecture: For every cardinal ${\displaystyle \lambda }$ there exists a cardinal ${\displaystyle \mu (\lambda )}$ such that If an AEC K with LS(K)<= ${\displaystyle \lambda }$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.[109][115]
• Shelah's categoricity conjecture for ${\displaystyle L_{\omega _{1},\omega }}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[109]
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[116]
• If the class of atomic models of a complete first order theory is categorical in the ${\displaystyle \aleph _{n}}$, is it categorical in every cardinal?[117][118]
• Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
• Kueker's conjecture[119]
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property?
• 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?[120]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[121]
• Generalized star height problem

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