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.
Determines the largest possible sum of a monotonic subsequence of a sequence of n distinct real numbers.
Concerns the union of a family of at most a constant times two to the n many sets each of size n, for large n.
Estimates the least possible maximum, over subsets of the first n integers, of the sum of a plus or minus one valued function over the pairs inside that subset.
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.
Asks whether a graph on n vertices with no empty or complete subgraph of size ten times the logarithm of n has an induced non-trivial (neither empty nor complete) regular subgraph of logarithmic size.
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 a graph on n vertices with more than n²/4 edges has a triangle to which nearly half of all vertices are joined twice; the Erdős–Faudree conjecture, disproved by Ma and Tang's construction with constant 2 − √(5/2).
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.
Asks whether a graph on n vertices with no empty or complete subgraph of logarithmic size has exponentially many pairwise non-isomorphic induced subgraphs.
Asks whether a graph on n vertices with each degree repeated at most twice and more than half of n distinct degrees has a large empty or complete subgraph.
Determines the smallest and largest measure of the set where a monic real polynomial with all roots real in minus one to one has absolute value below one.
The radius of the largest disc inside the region where a monic complex polynomial with all roots in the unit disc has absolute value below one.
The least area of the region where a monic polynomial with all roots from a fixed closed infinite set of complex numbers has absolute value below one.
Examines a polynomial whose roots all lie strictly inside the unit disc and the shape of the region where its absolute value is below one.
Asks what can be said about a closed set in the plane of transfinite diameter one that lies inside no closed disc of radius one.
Asks whether every monic non-constant complex polynomial has a line onto which the set where its absolute value is at most one projects to measure at most two.
Determines the infimum of the largest component boundary length of the region where a polynomial with all roots in the closed unit disc is below one.
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 set where a monic complex polynomial has absolute value below one must lie in a disc of radius two whenever that set is connected.
Examines a monic polynomial with m distinct roots and a threshold so small that the set where its absolute value is at most that threshold has m components.
Asks whether a monic polynomial with all roots of absolute value at most r below two has a component of diameter over two minus r where it is below one.
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.
Asks whether the sum over all n of one over two to the power n minus three is irrational.