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.
Asks for an asymptotic formula for the fewest vertices of a triangle-free graph with chromatic number k, and for a proof that consecutive values have ratio tending to one.
Estimates the number of edge-disjoint complete graphs needed to partition the edges of a graph on n vertices with more than n squared over 4 edges; open; Győri and Keszegh settle the K_4-free case up to about n squared / 16.
Asks whether, for r at least 3 and n at least rk, the most edges in an r-uniform hypergraph on n vertices with no k pairwise disjoint edges is the larger of the clique count and the star count.
Asks whether the Ramsey number for a complete graph on k vertices, divided by k times two to the k over two, tends to infinity.
Asks whether the off-diagonal Ramsey number R(k+1,k) exceeds the diagonal number R(k,k) by a constant factor in the limit.
Concerns graphs of chromatic number four in which deleting any edge drops the chromatic number to three.
Estimates the largest degree sum forced on a triangle in a graph on n vertices with more than n²/4 edges, and asks whether it is at least about 1.464 n; open, between Fan's 21n/16 and a construction's 2(√3 − 1)n + O(1).
Asks whether some positive c makes minimum degree above one minus c times two to the n force an n-dimensional hypercube in a graph on two to the n vertices.
The maximum product of all pairwise distances among complex numbers that are pairwise at most distance two apart, and whether a regular polygon is optimal.
Asks whether the sum over all n of one over t to the power n minus one is irrational for every rational t greater than one.
Concerns unitary perfect numbers, integers equal to the sum of their proper divisors d for which d and n divided by d are coprime.
Asks whether every k with the sum of divisors of n equal to k times n must be of order smaller than log log n.
Concerns the classification of primes into classes by repeatedly factoring p plus one, starting from primes whose only such factors are two and three.
Asks whether, for every k at least two, there are a prime and k consecutive intervals of integers whose products are each congruent to one modulo that prime.
Asks whether the number of Carmichael numbers up to x is x to the power one minus a quantity tending to zero.
Bounds the number f(n) of integers k with k times the sum of divisors of k equal to n, asking whether f(n) is at most n to a power o(1/log log n), perhaps even a power of log n.
Estimates the least n at least two k for which n minus i divides n choose k for all but one i below k.
Asks whether infinitely many primes are one plus a power of two times a prime, or one plus a power of two times a power of three times a prime.
Concerns the graph on n plane points that are pairwise at least distance one apart, with edges joining the pairs exactly distance one apart.
Asks whether every graph of chromatic number aleph one contains a countable subgraph that is infinitely vertex-connected.
Estimates the largest guaranteed number of points, among any n points in the plane, with no two at distance one, and whether it is at least n over four.
Studies the least n for which a given prime divides n factorial plus one, as a function of that prime.
Asks whether the number of composite numbers below x dividing n factorial plus one for some n is at most x to a power tending to zero.
Asks whether the m for which m factorial plus one has a prime factor not congruent to one modulo m, and the primes arising so (Pillai primes, counted among all primes), have densities, and what they are.
Asks whether n points in the plane with no three collinear always determine at least the floor of n/2 distinct distances, even as seen from a single point.