Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
99 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the count of totient values up to x doubles when x doubles, and whether that count has an asymptotic formula; the doubling limit is proved twice in Lean, and the release's asymptotic equivalent is a pending claim.
Compares the least prime congruent to one modulo n with the least integer whose Euler totient is divisible by n; a September 2026 forum claim answers the three questions no, no and yes, pending review.
Asks whether the set of integers avoiding a prescribed residue class pattern modulo each member of a given set of moduli always has a logarithmic density.
Asks whether the density of the multiples of a finite set up to m is under twice its density up to a smaller n at least the largest element; a counterexample claim of September 2026 is pending and unreviewed.
Asks whether the average of the squared gaps between consecutive integers divisible by no member of a sparse set of divisors tends to a finite limit.
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.
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 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.
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.
The best upper bound for the reciprocal sum of a set of integers up to N in which every number has at most r representations as a prime times a set element.
Asks for a proof bounding the Ramsey number of a large tree against a complete multipartite graph by a formula in its chromatic number and smallest class size.
Asks for the least constant c such that the Ramsey number of an odd cycle of length two k plus one against any m-edge graph without isolated vertices is at most c times m.
Characterizes the finite three-uniform hypergraphs that must appear in every three-uniform hypergraph with uncountable chromatic number.
Records the two-part Erdős–Pach–Pollack–Tuza diameter bound for connected graphs with no K_{2r} or K_{2r+1}: part (i) refuted in a refereed paper; part (ii) proved at r = 1, with pending claims that refute it.
Asks how large a triangle-free induced subgraph every K_4-free graph on n vertices must contain; the Erdős–Rogers problem, known to within a logarithmic factor in the refereed record and claimed to be sqrt(n log n) by a 2026 preprint.
Asks whether a function on the finite subsets of a set of size aleph_omega that never picks a member of its input (a set mapping) must have an infinite independent set.
Asks whether a hereditary family of finite graphs forcing monochromatic triangles under every finite number of colors holds the finite subgraphs of a graph forcing them under any infinite cardinal; a disproof is claimed.
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 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.
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, for large n, one residue class per prime up to n can be chosen so that every integer from 1 to n lies in at least two of them; open on the site, with three pending AI-assisted full claims of April and July 2026.
Asks whether a family of sets closed under subsets always has an element contained in as many members as the largest intersecting subfamily has sets.
Asks for an asymptotic formula for the shortest interval just above n with distinct integers, the kth divisible by k, for k up to n; known between n root log n over log log n and 1.74 n root log n, with a 2026 claim pending.
Asks for a function describing, for almost all n, the least integer that fails to divide the central binomial coefficient of n.