Catalogue

ProblemCatalogue

Open questions at the frontier of mathematics.

Open: 668
Catalogued: 1022
Attempted: 1

Showing 100 of 668 Search or filter to reach the rest.

Are there only finitely many powers of two whose ternary expansions contain no digit two?

Number theory / Numeral representations

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How far above one half must density be to force all pairwise sums of a large set?

Number theory / Additive combinatorics

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OPEN

Not attempted

How small an upper bound controls the powerful part of a product of consecutive integers?

Number theory / Powerful numbers

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OPEN

Not attempted

What conditions make a set an additive basis of prescribed order when summands must be distinct?

Number theory / Additive bases

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OPEN

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Is there an integer whose succeeding positions have divisor counts bounded linearly in their distance from it?

Number theory

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Determine how consecutive prime gaps increase, decrease, or remain equal, and how frequently each pattern occurs.

Number theory / Primes

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Do the positive odd integers not expressible as a prime plus two nonnegative powers of 2 have positive upper density?

Number theory / Additive bases / Primes

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What tradeoff relates the size of a subset whose pairwise sums avoid a given set to the number of forbidden sums?

Additive combinatorics

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What is the Ramsey number of a prescribed tree versus a complete multipartite graph?

Graph theory / Ramsey theory

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Do prime gaps, normalized by their average scale, have a limiting distribution?

Number theory / Primes

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Can several consecutive integer intervals each have product congruent to one modulo the same prime?

Number theory

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What structure is forced when the density of a sumset equals the sum of the densities of its two summand sets?

Number theory / Additive combinatorics

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What proportion of the edges of a high-dimensional hypercube can be retained without creating a quadrilateral?

Graph theory

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How large a subset with no isosceles triangle must every finite higher-dimensional point set contain?

Geometry / Distance geometry

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Can a dense integer set have the property that equal products always use the same number of factors?

Number theory

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Can the classical exponential lower bound for diagonal Ramsey numbers be improved by an unbounded factor?

Graph theory / Ramsey theory

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Can the divisor-difference sets of two distinct integers have arbitrarily many common values?

Number theory / Divisors

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OPEN

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Can one residue class per prescribed prime be chosen to cover each integer in a given interval at least twice?

Number theory

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Does high chromatic number and small clique number force two anticomplete vertex subsets each of high chromatic number?

Graph theory

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Which finite hypergraphs must occur in every three-uniform hypergraph of uncountable chromatic number?

Set theory / Graph theory / Hypergraphs / Chromatic number

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Does sufficiently high chromatic number force a triangle-free subgraph whose chromatic number reaches a prescribed uncountable cardinal?

Set theory / Chromatic number

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Does every fixed bipartite graph have an extremal function asymptotic to a constant times a rational power?

Graph theory / Turán numbers

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Can a dense graph be edge-colored with linearly many colors so that every four-cycle is rainbow?

Graph theory / Ramsey theory

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Chowla's cosine problem

Analysis

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How many levels in an exponential tower are needed to describe the growth of uniform-hypergraph Ramsey numbers?

Graph theory / Ramsey theory / Hypergraphs

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Must every permutation of the natural numbers contain a four-term arithmetic progression appearing in monotone order?

Arithmetic progressions

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How narrow an interval can contain all the distinct factors in a factorization of a factorial?

Number theory / Factorials

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How large an independent set is guaranteed by average degree in a graph excluding a prescribed clique?

Graph theory

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How many colors are needed for the graph joining integer-distance pairs of planar points in general position?

Graph theory / Chromatic number

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How many complete uniform hypergraphs suffice to partition the edges of an arbitrary uniform hypergraph?

Graph theory / Hypergraphs

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Near every integer, is there another integer whose least prime factor exceeds the square of their distance?

Number theory / Primes

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How much can the largest size of a Sidon set increase when the endpoint of its containing interval increases slightly?

Additive combinatorics / Sidon sets

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How fast do iterates of the sum-of-divisors function grow?

Number theory / Function iteration

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If at least half the vertices have sufficiently large degree, must a graph contain every tree of a prescribed order?

Graph theory

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ambiguous statement

Number theory

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Can the integer values of a polynomial form a Sidon sequence?

Number theory/ Powers / Sidon sets

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Existence of odd perfect numbers

Number theory

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Černý conjecture

Automata theory / Combinatorics

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For a square-root-size subset of an integer interval, how many distinct two-term sums can still lie in that interval?

Additive combinatorics

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How large a subset of an arbitrary finite real set can have all pairwise sums outside the original set?

Additive combinatorics

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What is the maximum edge count of a graph containing neither triangles nor four-cycles?

Graph theory / Turán numbers

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OPEN

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Must the decreasing sequence generated by the specified prime-factor rule contain a composite number?

Number theory

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Must two graphs of uncountable chromatic number have a common subgraph of large chromatic number?

Graph theory

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Can every finite set with distinct two-element sums be extended to a nearly optimal set with the same property?

Number theory / Sidon sets / Additive combinatorics

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Is there a dense subset of the plane in which every pairwise distance is rational?

Geometry / Distance geometry

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Successively sum the divisors greater than one of an integer. Which sums have not appeared before, and how small can the original integer representing a target be?

Number theory / Divisors

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Must a multicolored complete graph contain a vertex subset of the required size missing at least one color?

Graph theory

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How large are the maximum gaps among integers having a divisor in a prescribed size range?

Number theory / Divisors

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Can graphs of arbitrarily high chromatic number be constructed without changing the finite subgraph types that occur?

Graph theory / Chromatic number

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How fast does the mean endpoint distance of a random self-avoiding lattice walk grow with its length?

Geometry / Probability

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What is the smallest possible size of an additive complement of the squares that represents every sufficiently large integer?

Number theory / Additive bases

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Within the specified polynomial modulus region, how short a path joins the origin to the unit circle?

Analysis

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For sign-coefficient polynomials, must the maximum modulus on the unit circle exceed the square root of the number of terms by a fixed proportion?

Analysis / Polynomials

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How many distances determined by a finite planar point set must occur relatively infrequently?

Distance geometry

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What is the density of integers whose largest prime factor is smaller than that of the next integer?

Number theory

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Does every two-coloring of triples from a set of continuum cardinality contain a monochromatic subset of the prescribed order type?

Graph theory / Ramsey theory / Set theory

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Consecutive prime gap bound of 186

Analytic number theory

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Artin primitive root conjecture

Algebraic number theory / Analytic number theory

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Are there infinitely many consecutive positive integers with equal totients?

Number theory

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For a complex number outside the unit circle, can sums of consecutive powers approach zero exponentially fast?

Analysis

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How large a prime factor must a product of consecutive integer values of an irreducible polynomial have?

Number theory

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Brocard-Ramanujan conjecture

Number theory / Factorials

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Is there an infinite graph with no four-vertex clique that cannot be partitioned into countably many triangle-free parts?

Graph theory / Set theory

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If each prescribed local part of a set family has a two-point transversal, how many points suffice to meet the whole family?

Combinatorics

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How long can a consecutive-integer interval be if each term has a prime factor larger than its length?

Number theory

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What conditions on bases or exponent ensure the prescribed coprimality relations among like powers minus one?

Number theory

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Is the minimum-degree threshold forcing a quadrilateral monotone in the relevant size parameter?

Graph theory / Ramsey theory

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When do distinct-element sums of a sequence of floors of exponential values represent every sufficiently large integer?

Number theory / Complete sequences

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segmented numbers

Number theory

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How large a subset avoiding arithmetic progressions of a prescribed length must every finite integer set contain?

Additive combinatorics / Arithmetic progressions

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Herzog-Schönheim conjecture

Group theory / Covering systems

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Must a set of positive integers whose reciprocals have divergent sum contain arithmetic progressions of every finite length?

Number theory / Additive combinatorics / Arithmetic progressions

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Must every convex polygon have a vertex determining about half as many distinct distances as the total number of vertices?

Geometry / Convexity / Distance geometry

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Near every integer, is there a composite integer whose least prime factor exceeds the square of their distance?

Number theory / Primes

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Among forbidden graphs with fixed vertex and edge counts, which minimizes the extremal function? Is this minimum strictly monotone in the forbidden graph's edge count?

Graph theory / Turán numbers

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What extremal lower bounds can be constructed for uniform hypergraphs excluding a prescribed complete multipartite hypergraph?

Hypergraphs / Turán numbers

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What is the minimum host edge count forcing a monochromatic prescribed balanced complete bipartite graph under every two-coloring?

Graph theory / Ramsey theory

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Under the stated conditions, is every infinite subsum of reciprocals of powers of two minus one irrational?

Irrationality

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Are infinitely many central binomial coefficients coprime to 105, hence divisible by none of 3, 5, and 7?

Number theory / Binomial coefficients / Numeral representations

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For a fixed integer, how large is the reciprocal sum of its positive differences from preceding primes, and how does it vary?

Number theory / Primes

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Must a graph of uncountable chromatic number contain a countable subgraph that remains connected after every finite vertex deletion?

Graph theory / Set theory / Chromatic number

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Can every finite subgraph of a graph with uncountable chromatic number be made bipartite by deleting very few edges?

Graph theory / Chromatic number / Set theory

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Can a pairwise coprime integer sequence leave relatively small gaps after all its multiples are excluded?

Number theory

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If every prescribed-size local edge family in a uniform hypergraph has a common vertex, how many vertices suffice to cover all edges?

Graph theory

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What conditions ensure uniformly bounded gaps between differences occurring repeatedly in an integer set?

Number theory

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Decimal normality of pi

Number theory / Dynamical systems

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How much can the required order differ between additive bases allowing at most a prescribed number of summands and those requiring exactly that number?

Number theory / Additive bases

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Is there an integer such that distinct-prime-factor counts at all preceding positions are uniformly controlled by distance from it?

Number theory

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When does iteration of the map sending an integer to itself plus its totient eventually follow a doubling pattern?

Number theory / Function iteration

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Does sufficiently high minimum degree force a spanning hypercube?

Graph theory

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minimum overlap problem

Number theory / Additive combinatorics

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How close to a given integer can one find a composite number whose least prime factor satisfies the prescribed size conditions?

Number theory / Primes

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Can the mean square of representation counts in an additive basis remain bounded?

Additive combinatorics / Additive bases

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For a given number of points and distance constraints, how many noncongruent configurations attain the minimum diameter?

Geometry / Distance geometry

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Which monochromatic triangles or order types are forced by multicolorings on the ordinal square of the first uncountable ordinal?

Set theory / Ramsey theory

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Elliott–Halberstam conjecture

Analytic number theory

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How many planar points in general position can have all pairwise distances integral?

Geometry / Distance geometry

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Not attempted

How does the size Ramsey number grow with the target graph's average degree?

Graph theory / Ramsey theory

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How strong a lower bound on a graph's Ramsey number follows from its chromatic number?

Graph theory / Ramsey theory

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How does the minimum number of edges needed to make a graph triangle-free compare with its maximum number of edge-disjoint triangles?

Graph theory

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Not attempted

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