Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
516 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
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.
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.
Estimates the longest run of pairwise distinct consecutive prime gaps starting at an index below x, asking whether it exceeds a power of log x and whether it is o(log x).
Asks whether the smallest even number missing from the first x prime gaps tends to infinity, and whether it grows faster than log x.
Asks about the arithmetic of the primorials, the products of the first k primes.
Asks whether the number of primes up to x plus y is at most the number up to x plus the number up to y, for all large x and y.
Estimates, for k at least three, the largest reciprocal sum of a set of integers up to N with no k members sharing the same pairwise least common multiple.
Estimates the least number of subsets of the integers up to n that forces a sunflower of size k, meaning k of them with equal pairwise intersections.
Estimates the shortest interval length that always contains distinct integers, one divisible by each prime up to n.
Estimates the largest set of integers up to N with at most one number having more than one representation as a sum of two members.
Estimates, for k at least three, how far above N a subset of the integers up to 2N must be to force k integers whose pairwise sums all lie in the set; open, with the thresholds 1 and 3 for k = 3 and 4 and bounded for k = 5.
How long the game in which two players alternately add integers up to n to a shared set free of divisibility between members can be guaranteed to last, and whether it lasts at least εn or (1-ε)n/2 moves.
Asks whether, for every positive epsilon, some k makes the number of windows of k consecutive members with least common multiple below X less than X to epsilon.
Determines how slowly an infinite set of naturals can grow while its sets of sums of r distinct members stay disjoint for different r.
Compares f(n), the sum over the primes p dividing n of the largest power of p not exceeding n, with F(n), the largest sum of distinct pairwise coprime integers from 2 to n built from those primes.
Estimates the largest sum of a set of pairwise coprime integers up to n, and asks whether the best such set must contain a number with at least k prime factors.
Asks whether, for every k, there are k integers whose sets of differences of complementary factor pairs share at least k common values.
Asks whether, for each fixed positive epsilon, every large n has only boundedly many divisors just above the square root of n.
Asks whether there is an absolute bound on the number of divisors of a large n lying within a constant times the fourth root of n above its square root.
Concerns the number of prime factors of n plus k that exceed k, that is, those dividing none of the earlier terms of the interval.