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 whether infinitely many n, or any n above one hundred and five, make n minus every power of two greater than one and less than n prime.
Estimates how many multiples of at least one of finitely many given primes every interval of k consecutive positive integers must contain.
Asks a question about two increasing sets of positive integers whose nth elements have ratio tending to one.
Asks a question about essential components, sets whose sum with any other set of Schnirelmann density strictly between zero and one raises that density.
Asks whether, for every number between zero and one, some ring or field of real numbers has exactly that Hausdorff dimension; yes under the continuum hypothesis by Mauldin's theorem, so the existence cannot be refuted in ZFC.
Asks for the typical structure of the graph left by repeated uniform random triangle removal from K_n and whether its edge count has order n^{3/2}; the sharp constant is formally verified, and the typical structure is open.
Asks a question about the chromatic number of a random graph on n vertices in which each edge appears independently with probability one half.
Asks for the most edges an r-graph on n vertices can have with no k vertices spanning s edges, the Brown-Erdős-Sós problem; the conjecture's s = 3 case and large-uniformity linear form are proved; 3-graphs with k = s + 3, s ≥ 4, open.
Asks whether Erdős's 1964 upper exponent for the Turán number of the complete t-partite t-uniform hypergraph with r vertices per class is attained up to o(1); known only for t = 2 and r at most 3.
Asks whether there is a constant greater than one for which a stated combinatorial property holds.
Asks whether the number of groups of order n is at most the number of groups of order two to the m whenever n is at most two to the m.
Asks for an asymptotic formula for the number of subgroups of the symmetric group on n letters, and for statistical results on their orders.
Asks for a statistical description of the arithmetic structure of the orders of subgroups of the symmetric group on n letters.
Asks about a partition problem for infinite cardinals, coloring r-element sets when the parts are given by a sequence of prescribed cardinals; open under the Erdős–Hajnal list's conditions gamma at least 2 and every kappa_alpha above r, which exclude the failures of the site's wording.
Asks for a proof of a negative partition relation for pairs at the successor of the omega-th infinite cardinal, without the generalized continuum hypothesis.
Asks whether the square of the first uncountable ordinal fails a partition relation for pairs into itself and a triangle.
Asks whether it is consistent that the second uncountable ordinal has the two-color partition property for pairs for every smaller ordinal.
Asks whether the square of the first uncountable ordinal has a partition property for pairs giving a large ordinal or a triangle, for each finite color count.
Asks whether several partition relations for pairs of small uncountable ordinals hold, or are consistent, under the generalized continuum hypothesis.
Asks whether, under the generalized continuum hypothesis, a set mapping on omega_{omega+1} with values of size at most aleph_omega and pairwise intersections below aleph_omega must have a free set of size aleph_{omega+1}.
Asks whether some K_4-free graph forces a monochromatic triangle, and whether some K_{aleph_1}-free graph forces a monochromatic K_{aleph_0}, in every edge coloring with countably many colors.
Asks whether, for each uncountable cardinal kappa, some cardinal lambda makes every graph of chromatic number lambda contain a triangle-free subgraph of chromatic number kappa.
Asks whether every graph of chromatic number aleph_1 has an edge coloring with aleph_1 colors such that every countable vertex coloring has a class containing edges of all colors.
Determines the least number of vertices d such that forbidding all r-uniform hypergraphs with d vertices and e edges forces subquadratically many edges.
Asks whether the least prime not dividing the product of the next roughly log n integers after n is below (1-c)(log n)^2 for some c>0 and all large n; only the trivial bound (1+o(1))(log n)^2 is known.