Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the count of integers up to x that are sums of two squarefull numbers is asymptotic to a constant times x over the square root of log x.
Estimates the fewest points in d-dimensional space guaranteeing at least n distinct distances, and whether dividing it by d to the n minus one has a limit.
Asks whether, for each k at least 3, some finite planar set has every two-coloring giving a line whose at least k points of the set share one color.
Asks whether every graph with chromatic number four and no complete graph on four vertices contains an odd cycle with at least two diagonals; Erdős first asked for one diagonal, which Larson proved in 1979.
Determines the largest f so that a graph whose every m-vertex subgraph is an r-colorable graph plus at most f edges has chromatic number at most r plus one.
Asks whether the gaps between consecutive finite sums of distinct powers of q tend to zero for every q slightly above one; proved by Erdős and Komornik (1998), Akiyama and Komornik (2013) and Feng (2016), each for a range of q.
Concerns the non-commuting graph of a group, whose vertices are the group elements and whose edges join pairs that do not commute.
Asks whether the sum of the ratios of consecutive divisors of n minus one, each raised to a power alpha above one, has bounded limit inferior over all n.
Asks how fast a sequence must increase if, for every n, only finitely many members a make n+a squarefree (property P), or if for infinitely many n every member a<n makes n+a squarefree (property Q).
Asks for the anti-Ramsey numbers of cycles and paths, the most colors on the edges of the complete graph on n vertices without a rainbow copy: an asymptotic formula for cycles and an exact formula for paths.
Asks whether, for given gap bounds and k at least 3, every sufficiently lacunary sequence misses the k-fold sumset of some sequence with those gaps.
Concerns real polynomials of degree n whose roots are all real and form an arithmetic progression.
The linear-length asymptotic-path conjecture fails at every finite order, even for functions growing arbitrarily close to the logarithmic-square threshold that guarantees radial paths.
Asks whether some meromorphic or entire function has, for every two distinct values, the ratio of their solution counts in growing discs unbounded.
Concerns non-constant entire functions for which the set where the modulus exceeds some constant has finite measure.
Asks whether a family of entire functions taking at most m distinct values at each point has cardinality at most m, for m between countable and continuum.
Asks whether circles in the plane that no disjoint line separates can always be covered by a single circle whose radius is the sum of their radii.
Compares the Boolean algebra of sets of integers modulo density zero with the Boolean algebra of sets modulo logarithmic density zero.
Asks whether a square and a circle of the same area can be cut into finitely many congruent pieces.
Asks whether a real function with twice its value at a point at most the sum of its values at two later equally spaced points must be monotonic.
Asks whether a function additive for almost all pairs of reals must agree almost everywhere with a function that is additive for all pairs.
Asks whether real n-dimensional space splits into countably many sets in each of which all pairwise distances are distinct.
Asks whether every two-coloring of a product of three sets of size aleph one contains a monochromatic product of three countably infinite subsets.
Asks a question about how large the fundamental Lagrange interpolation polynomials for nodes in the interval from minus one to one can be.
Asks for the extreme behavior of sums of the fundamental Lagrange interpolation polynomials built from nodes in the interval from minus one to one.