Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
516 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Estimates how large a monochromatic family closed under unions and intersections must exist in every two-coloring of the subsets of the first n integers; open, with nothing beyond the trivial chain bound proved.
Asks whether the count of integers just above n whose largest prime factor exceeds k follows the prediction given by the Dickman function.
Estimates the least density of monochromatic k-term arithmetic progressions forced in every two-coloring of the first n integers.
Asks whether every infinite Sidon set has counting function o((x/log x)^(1/2)) along a subsequence, and whether some infinite Sidon set keeps it at least x^(1/2)/(log x)^c for some c > 0.
Concerns the possible growth of the number of representations of n as a sum of r elements of a set of natural numbers.
For a set in which every positive integer n is uniquely a difference a_n - b_n of two members, asks how fast a_n/n must grow, where a_n is the larger member of the representation of n.
Asks for primes below x with bounded reciprocal sum together with residues covering every integer less than x.
Asks whether the maximum over k of the number of distinct prime factors of n plus k times log log k over log k tends to infinity as n grows.
Estimates the smallest largest element, and smallest average, of k integers missing a residue class modulo every prime; the largest element lies between half and once k log k, the average between a quarter and half of it; open.
Asks whether the set of cubes up to N cubed contains a Sidon set, with all pairwise sums distinct, of size proportional to N.
Estimates the largest subset with no isosceles triangle guaranteed in any n points in d dimensions, in particular whether in the plane it is below a power of n.
Estimates, for fixed dimension d, the largest number of points with all distances distinct that must lie inside every set of n points in d-dimensional space.
Asks whether one shift n making every n plus a term of a fast-growing sequence prime forces infinitely many, with squarefree and doubly exponential variants; three questions answered no, three stay open.
Asks whether a set of pairwise coprime integers below n has the sum of one over n minus a at most the sum of reciprocals of primes below n plus a constant.
Asks whether, under GCH, the successor of aleph_{omega_{omega+1}} fails the partition relation for triples whose first target is that cardinal and whose other countably many targets are 4.
Asks whether, for a sequence on the circle, the de Bruijn–Erdős constants for the largest and smallest sums of r consecutive gaps and for their ratio deviate from their trivial values by more than any constant over r as r grows; a 2026 preprint claims all three parts for distinct points, unrefereed and unreviewed.