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 bounded sets of outer measure below one assigned to the reals leave an infinite set no member of which lies in another's set (independent of ZFC), and whether closed sets of measure below one leave three (proved).
Estimates the largest size of a set of points in n-dimensional space realizing only two distinct pairwise distances; settled to leading order: n^2/2 + O(n).
Determines the largest size of a set of points in d-dimensional space in which every three points form an isosceles triangle.
Determines the largest angle that is guaranteed to appear among some three points of every set of n points in the plane.
Asks whether every set of diameter one in n-dimensional space splits into at most n plus one pieces of smaller diameter.
Determines the least number of circles determined by n points of the plane, not all on one circle or one line (Elliott's reading); known for n > 393 since Purdy and Smith's correction; a 2026 claim of every value is pending.
Estimates the smallest area such that every set of n points in the unit disk contains three points forming a triangle of at most that area.
Determines the fewest colors needed to color the plane so that no two points at distance one share a color.
Asks whether the set where a monic nonconstant complex polynomial has modulus at most one can be covered by circles whose radii sum to at most two.
Asks whether every set of N positive integers admits an angle where the sum of the cosines of its members times that angle is below a negative constant times root N; the site's wording over all integers fails at sets containing zero.
Asks whether the set where a monic polynomial has modulus below one has boundedly many components of diameter above a fixed constant, whatever the degree.
Asks whether the mean absolute value of the exponential sum over a set of N integers is at least of order the logarithm of N.
Determines the largest limit inferior of the ratio of the maximal power series term of a transcendental entire function to its maximum modulus on radius r.
Asks whether every transcendental entire function has a path to infinity on which it outgrows every power of the variable, how long such a path must be, and whether it can outgrow a fixed function of the maximum modulus.
Asks whether every entire nonpolynomial function has a rectifiable path to infinity along which the integral of any negative power of its modulus is finite.
Asks whether an entire function of finite order with very sparse exponents has minimum modulus whose logarithm matches that of its maximum modulus.
Asks whether an entire power series whose exponents grow faster than linearly in the index must take every complex value infinitely often.
Asks whether every two-coloring of the complete graph on n vertices admits root n monochromatic paths of one color covering all vertices; proved for n above 20 to the 40th, the remaining n claimed in an unrefereed 2026 preprint.
Asks whether the largest modulus among the first n power sums of complex numbers, one of which is one, is bounded below by an absolute positive constant.
Asks whether Rademacher random multiplicative sums over sqrt(N log log N) almost surely have a positive constant limit superior; a Lean proof built and audited here makes the ratio tend to 0, so the answer is no.
Asks whether, for a random polynomial of degree n with independent sign coefficients, the count of real roots divided by log n tends almost surely to two over pi.
Concerns the behavior of a random polynomial of degree n whose coefficients are chosen independently and uniformly from plus one and minus one.
Concerns the behavior of a random polynomial of degree n whose coefficients are chosen independently and uniformly from plus one and minus one.
The order of magnitude, for almost every real number in the unit interval, of the maximum on minus one to one of the polynomial built from its binary digits.
Asks whether almost all degree n polynomials with coefficients plus or minus one dip below absolute value one on the unit circle, and how small that minimum is.