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 the number of points in general position in k-dimensional space needed to guarantee n of them in convex position grows exponentially in n.
Asks whether the least possible number of distinct distances from the k-th of n planar points, in units of root n, grows with k; Erdős's first guess, that it is unbounded already at k = 3, fails by a construction of Elekes.
Asks whether n points in the plane can take almost n different values among the counts of distinct distances from each point to the others; yes by a Lean proof certified by Conjectures.io, unrefereed, kernel-checked by that site.
Estimates how many distinct distances to other points some point must have, among n points in the plane with no four of them on a circle.
Asks whether n planar points, with no circle centered at one of them holding three others, determine more than half of n distinct distances by a constant factor.
Asks whether every set of positive upper density contains, after some shift, all pairwise sums of distinct members of an infinite subset.
Asks whether n planar points forming no isosceles triangle must determine a number of distinct distances that grows faster than a constant times n.
Asks whether every subset of the N by N grid of positive density contains the four vertices of a square, once N is large enough.
Asks whether n planar points can have every four of them determining at least three distances while the total number of distinct distances is far below n.
Asks whether the n vertices of a convex polyhedron in space always determine nearly half of n distinct distances.
Asks whether two sets of n planar points can have fewer than n over the square root of the logarithm of n distinct distances between the two sets.
Asks whether n points at mutual distance at least one have at most f(t) distances at most t, f(t) the triangular lattice's count; garbled as worded, it fails under every counting reading, and a disproof is claimed.
Asks whether the least prime not dividing the product of k consecutive integers above n is at most about the logarithm of n, for fixed k and large n.
Concerns families of subsets of the first n integers, each of size above a constant times the square root of n, with any two sharing at most one element.
Concerns pairwise balanced designs on the first n integers, families of sets in which every pair of distinct elements lies in exactly one set.
Asks whether every subgraph of the n-dimensional hypercube with a positive fraction of its edges contains a six-cycle, once n is large enough.
Asks whether the exponent governing the largest clique forced when every p vertices span at least q edges is strictly increasing in q; open, with the 1997 source's endpoint bounds and a disputed upper bound at the top.
Asks whether the number of incongruent n-point planar sets maximizing the number of unit distances tends to infinity, and exceeds one for every n above three.
Bounds how many lines can pass through at least k, or through exactly k, of n given points in the plane.
Asks whether n points in d-dimensional space whose pairwise distances all differ by at least one must have diameter at least (1 + o(1)) n squared.
Concerns sequences of Lagrange interpolation polynomials built on nodes in the interval from minus one to one and how they behave as the degree grows.
Asks whether a product of at least four positive terms in a primitive arithmetic progression, with gcd of initial term and difference one, can be a perfect power.
Asks whether the sum of the ratios of consecutive divisors of n tends to infinity for almost all n, and seeks an asymptotic formula for its average.
Asks whether x to the power x times y to the power y equals z to the power z has integer solutions with x, y and z all greater than one.
Asks which sets of integers are locally periodic, in that membership up to n is unchanged by some shift, for sums of two squares and other examples.