Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
1,221 problems
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether every graph of chromatic number aleph_1 has an edge coloring with aleph_1 colors such that every countable vertex coloring has a class containing edges of all colors.
Concerns the families of three-uniform hypergraphs of a given chromatic number that avoid a fixed finite three-uniform hypergraph.
Determines the least number of vertices d such that forbidding all r-uniform hypergraphs with d vertices and e edges forces subquadratically many edges.
Estimates the least size of a random subset of an abelian group of order N whose subset sums hit every group element nearly equally often.
Asks whether, for each positive epsilon, boundedly many inverses of integers up to p^epsilon represent every residue modulo any prime p; proved by Shparlinski, Croot and Glibichuk, whose bound has order epsilon^(-2).
Asks whether the least prime not dividing the product of the next roughly log n integers after n is below (1-c)(log n)^2 for some c>0 and all large n; only the trivial bound (1+o(1))(log n)^2 is known.
Estimates the largest edge counts for connected graphs on n vertices whose Ramsey number against a triangle equals two n minus one; a 1996 preprint claims a linear threshold for all such graphs, a no to the closing question.
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.
Asks whether a dense set of integers has a k-term arithmetic progression whose common difference is a difference of two members of any large enough set.
Estimates the least density of monochromatic k-term arithmetic progressions forced in every two-coloring of the first n integers.
Asks whether every coloring of the integers with finitely many colors contains k primes in arithmetic progression all of the same color.
Concerns covering systems with distinct moduli that are minimal, in that no proper subsystem still covers every integer.
Concerns sets of distinct moduli that can cover the integers by some choice of residues but have no proper subset that can.
Asks for the size of the supremum of reciprocal sums over finite disjoint congruence families with distinct moduli greater than m; the supremum is known to logarithmic scale.
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.
Asks whether, for a set A of natural numbers and a positive non-decreasing g, the set of n at which the number of representations of n as a sum of two elements of A equals g(n) always has lower density 0, and upper density below some c < 1.
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.
Concerns sets of real numbers of infinite measure in which no ratio of two distinct elements is an integer.
Asks whether the sum of one over a times log a over a primitive set of integers all at least x is at most one plus a quantity tending to zero.
Asks whether, for a set of positive measure and almost every positive x, every large integer multiple of x lies in some integer dilate of the set.
Asks whether every two-coloring of the natural numbers admits an infinite set all of whose sums of products of distinct members share one color; no, by a 1995 theorem of Smith, as the site's thread deduced in 2026.
Asks whether every two-coloring of the natural numbers admits an infinite set whose pairwise sums, doubles included, share one color (Owings's question); open on the site, false for three colors, claimed for two.
Asks for primes below x with bounded reciprocal sum together with residues covering every integer less than x.