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 n disjoint triangles joined by a Hamiltonian cycle on their 3n vertices always give a 3-colorable graph; proved by Fleischner and Stiebitz.
Asks whether the squares are Ramsey 2-complete, so that any 2-coloring of them leaves all large integers as sums of distinct same-colored squares.
Bounds the largest set of integers up to N in which the product of any two members is never squarefree.
Asks whether, for each constant C, the sums of distinct products of powers of two and three lying within a factor C of each other have density zero.
Asks what follows for an infinite plane set in which every n of its points contain at least a fixed proportion with no three collinear.
Asks what follows for an infinite set of naturals in which every n of its members contain a fixed proportion forming no three-term arithmetic progression.
Asks whether some bounded r makes the integers of the form a power of two plus a number with at most r prime divisors have density at least one minus epsilon.
Estimates the largest reciprocal sum, over the logarithm of N, of a set of integers up to N in which no member equals another member times a factor whose prime factors all exceed the smaller member.
Asks whether the density of the integers n for which a given t is a sum of distinct divisors of n is asymptotic to a constant over a power of log t; false by a Lean disproof the bounty site Conjectures.io certified in 2026.
Asks how many Sidon subsets of the integers up to N there are, compared with two to the power of the largest Sidon set size.
Asks whether the number of maximal Sidon subsets of the integers up to N is below two to the power o(square root of N), and whether it exceeds two to the power N^c for some c > 0.
Asks whether, for r at least 2, the largest sets up to N with at most r representations of each sum, and of each positive difference, have different square-root constants, and whether the difference constant is the smaller.
Asks whether every set of integers up to N of size just above five eighths of N contains three members whose three pairwise sums also lie in the set; proved in a 2026 preprint, developed with GPT-5.5 Pro, that the site accepted.
Asks whether a set of integers up to N in which no sum of consecutive members lies in the set has size at most half of N plus a constant; false, by Freud's 1993 construction of density 19/36.
Asks whether an additive basis of order two whose representation counts tend to infinity must contain a minimal such basis.
Asks whether the union of two disjoint additive bases of order two must contain a minimal additive basis of order two.
Asks whether an additive basis of order two whose representation counts tend to infinity can be split into two disjoint additive bases of order two.
Estimates the largest set of integers up to N whose sets of sums of r distinct members are disjoint for distinct r, and whether it nears two root N.
Estimates the number of maximal sum-free subsets of the integers up to n and whether it is o(2^(n/2)); yes, and the count is a residue-dependent constant times two to the power of n over 4.
Asks whether the integers representable as sums of k or fewer distinct members of an additive basis of order k have bounded gaps.
The size of the largest subset of one to n whose nonempty subset sums form a set in which no element divides another.
Asks whether the sum of one over all differences of divisors of n is bounded by a constant times one plus the sum of one over consecutive divisor gaps.
The size of the largest subset of one to n in which any four elements with square product must pair off so the outer product equals the inner product.
Asks whether the integers can be finitely colored with no two of one color differing by a term of a given lacunary sequence; proved, with Peres and Schlag's bound of order (1/epsilon) log(1/epsilon) colors as the best known.
Asks whether every large triangle-free graph on one to n contains three pairwise nonadjacent numbers of the form a, b and a plus b.