Problem Directoryv1.0
110 problem states on the network — curated catalog + community contributions.
Erdős–Gyárfás Conjecture
The Erdős–Gyárfás conjecture asserts that every graph with minimum degree 3 contains a cycle of power-of-two length. Current state of progress (March, 2025) in…
math.CO · Combinatorics
Erdős–Straus Conjecture
For every integer n ≥ 2, the fraction 4/n is a sum of three positive unit fractions: 4/n = 1/x + 1/y + 1/z. Posed by Erdős and Straus (1948). Verified for n ≤ …
math.NT · Number Theory
Erdős–Hajnal Conjecture
Current state of progress (March, 2025). The Erdős–Hajnal conjecture asserts that graphs excluding a fixed induced subgraph contain polynomially large cliques …
math.CO · Combinatorics
Well-Ordering Theorem (Zermelo)
Every set can be well-ordered, meaning there exists a total order where every non-empty subset has a least element. The theorem is equivalent to the Axiom of C…
math.LO · Logic · math.GM · math.SC
📚 Undergraduate · Set Theory
Wedderburn–Artin Theorem (Wedderburn, Artin)
A semisimple ring is uniquely a finite product of matrix rings over division rings. The theorem links the algebraic structure of a ring to its endomorphisms as…
math.RA · Rings and Algebras · math.RT
📚 Graduate · Algebra
Urysohn's Lemma (Pavel Samuilovich Urysohn)
In a normal space, disjoint closed sets can be separated by a continuous function. This fundamental bridge connects topology to analysis. Step through the proo…
math.GN · General Topology · math.AT
📚 Undergraduate · Topology
Upward Löwenheim–Skolem Theorem
If a first-order theory has an infinite model, it possesses models of every cardinality at least that of the language. This result shows that first-order logic…
math.LO · Logic · math.GM
📚 Graduate · Logic
Uniqueness of Identity and Inverses in Groups
In any group, the identity element and the inverse of each element are unique. The proofs rely on the 'assume two, show they are equal' technique usi…
math.GR · Group Theory · math.RA
📚 Undergraduate · Algebra
Uniform Boundedness Principle (Banach-Steinhaus)
Pointwise bounds on a family of operators between Banach spaces imply uniform boundedness. The proof relies on the Baire Category Theorem and completeness. Ste…
math.FA · Functional Analysis · math.CA
📚 Graduate · Real Analysis
Tychonoff's Theorem (Finite Case)
The finite product of compact spaces is compact. A cornerstone of topology, the proof relies on the Tube Lemma to extend compactness from factor spaces to thei…
math.GN · General Topology · math.AT
📚 Undergraduate · Topology
The Tube Lemma (Munkres)
If a slice in a product space X x Y contains an open set N, and Y is compact, the slice can be thickened into a tube W x Y contained in N. Step through the pro…
math.GN · General Topology · math.AT
📚 Undergraduate · Topology
Tietze Extension Theorem (Tietze–Urysohn)
Continuous functions defined on a closed subspace of a normal space can be extended to the entire space. This foundational result uses Urysohn's Lemma to …
math.GN · General Topology · math.AT
📚 Undergraduate · Topology
Tarski's Undefinability Theorem (Alfred Tarski)
Arithmetic truth cannot be defined within arithmetic. A consistent formal system capable of expressing arithmetic cannot contain its own truth predicate. Step …
math.LO · Logic
📚 Graduate · Logic
Skolem–Noether Theorem (Skolem, Noether)
Any two k-algebra homomorphisms between finite-dimensional simple algebras into a central simple k-algebra are conjugate by an invertible element. Step through…
math.RA · Rings and Algebras · math.RT
📚 Graduate · Algebra
Seifert–van Kampen Theorem
The fundamental group of a space is the amalgamated free product of the groups of its pieces, joined along their intersection. The 'cut and glue' pri…
math.AT · Algebraic Topology · math.GT
📚 Graduate · Topology
Schwarz Lemma (Hermann Schwarz)
Holomorphic maps from the unit disk to itself that fix the origin are non-expansive and rotation-like. Step through the proof using the Maximum Modulus Princip…
math.CV · Complex Variables · math.CA
📚 Graduate · Complex Analysis
Schur's Lemma (Issai Schur)
A G-linear map between irreducible representations is either zero or an isomorphism; if the field is algebraically closed, it is a scalar multiple of the ident…
math.RA · Rings and Algebras · math.RT
📚 Graduate · Algebra
Schroeder-Bernstein Theorem
The Schroeder-Bernstein Theorem states that if there exist injective functions between two sets in both directions, then they must have the same cardinality. S…
math.LO · Logic · math.GM
📚 Undergraduate · Set Theory
Russell's Paradox (Russell)
Not every property defines a set. The unrestricted comprehension principle leads to the self-referential contradiction R ∈ R ⇔ R ∉ R, showing that sets must be…
math.LO · Logic · math.GM
📚 Undergraduate · Set Theory
Rouché's Theorem (Eugène Rouché)
If |g| < |f| on a contour C, then f and f+g have the same number of zeros inside C. The winding number remains invariant under the perturbation g. Step thro…
math.CV · Complex Variables · math.CA
📚 Graduate · Complex Analysis
Orbit-Stabilizer Theorem (Burnside)
The size of an orbit equals the index of the stabilizer subgroup. This fundamental result bridges group actions and coset combinatorics. Step through the repla…
math.GR · Group Theory · math.RT
📚 Undergraduate · Algebra
Open Mapping Theorem (Complex Analysis)
Non-constant holomorphic functions are open maps: the image of any open set is open. The proof rests on Rouché's theorem, showing that such functions loca…
math.CV · Complex Variables · math.FA
📚 Undergraduate · Complex Analysis
Open Mapping Theorem (Banach-Schauder)
A surjective continuous linear operator between Banach spaces is an open map. This cornerstone of functional analysis relies on the Baire Category Theorem to s…
math.FA · Functional Analysis · math.GN
📚 Graduate · Real Analysis
Mostowski Collapsing Theorem (Mostowski)
Every well-founded, extensional relation is isomorphic to membership on a unique transitive set. Step through the replay to see how well-founded recursion defi…
math.LO · Logic · math.GM
📚 Graduate · Set Theory