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 there are infinitely many n for which the number of divisors of n plus k is at most a constant times k for every k at least one.
Estimates how many points in general position in the plane force k of them whose triples all determine circles of distinct radii.
Asks whether, for every integer a, there are infinitely many n whose Euler totient divides n plus a.
Asks whether the number of ways to write n as a sum of two cubes is at most a power of the logarithm of n.
Asks whether there are infinitely many amicable pairs, and whether the count of them up to x is at least x to the power one minus a small amount.
Estimates the least number of distinct radii among circles through three of n points in the plane, with no three collinear and no four concyclic.
Records Alon's counterexamples to the complete-hypergraph edge benchmark, while separating the defective equality wording and the open r=3 case.
Records the Erdős–Lovász exponential vertex-degree bound and an explicit constant that covers every uniformity r at least two.
Asks whether there is a three-critical three-uniform hypergraph in which every vertex has degree at least seven.
Asks whether some k above two lets the k-element subsets of the integers up to two k get k plus one colors so every k plus one of them sees all colors.
Separates the false vertex-bound question from the unresolved linear intersection question and repairs the site's chromatic-number gloss.
Asks for the set A_3 of densities alpha such that 3-uniform hypergraphs of limiting density above alpha must have growing subgraphs of density above some fixed beta > alpha, while limiting density at least alpha need not.
Estimates the least number of distinct convex subsets determined by n points in the plane with no three collinear, in particular whether a certain limit exists.
Asks whether an increasing integer sequence in which no term is a sum of consecutive earlier terms must have terms growing faster than linearly at times.
Determines how fast the largest quasi-Sidon subset of the integers up to N grows, where a set is quasi-Sidon if its sumset is nearly as large as possible.
Estimates the least length of a run of integers just above n that contains a subset whose product with n is a perfect square; Erdős's original question, whether the n with t_n at least n^{1-o(1)} have density zero, has the answer yes.
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 the largest set of integers up to N with no two members whose product plus one is squarefree is those congruent to seven mod twenty-five.
Asks whether, for every t at least one, some value is taken by exactly t binomial coefficients with the lower index between one and half the upper.
Asks whether two distinct integers can agree in prime factors, with their successors also agreeing and the next integers after those agreeing too.