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.
Asks whether n plane points with no five on a line determine only a negligible fraction of n squared lines containing exactly four points.
Estimates the largest number of collinear points forced when n plane points admit a constant times n squared lines with more than three points each.
Asks whether the number of incongruent n-point plane sets of minimum distance one and least possible diameter grows without bound.
Asks whether n points in the plane lie on only a negligible fraction of n squared distinct unit circles containing three or more of them.
Asks whether disjoint plane sets of n and n minus 3 points, the first not all collinear, admit a line meeting two points of the first and none of the second.
Asks whether the largest total side length of interior-disjoint squares packed in the unit square equals k when there are k squared plus one squares.
Asks whether two to the power n minus 2, plus one, points in the plane with no three collinear are always enough to force a convex n-gon.
Asks whether, for every girth bound at least 4 and every k, large enough chromatic number forces a subgraph of that girth with chromatic number at least k; disproved at girth 5 and k = 7 by a Lean counterexample family.
Shows that any set of natural numbers with positive upper density contains the sumset of two infinite sets.
Asks whether one function F(n) bounds, for all large n, the order of a smallest subgraph of chromatic number n in every graph of chromatic number aleph-one.
Asks how many edge deletions make an n-vertex subgraph bipartite, and whether that number grows faster than n for graphs of uncountable chromatic number.
Determines the fewest vertices forcing every directed graph to contain an independent set of size n or a transitive tournament of size m; open, known exactly when n = 1, m <= 2, n = 2 and m <= 6, or m = 3 and n <= 5.
Asks whether a bipartite graph has Turan number O(n^(3/2)) exactly when it has no induced subgraph of minimum degree at least three.
Asks whether the curve where a monic degree n complex polynomial has absolute value one is longest for the polynomial z to the n minus one.
Asks whether a monic degree n polynomial whose set of modulus at most one is connected has derivative at most (1/2+o(1))n^2 there; Eremenko and Lempert proved it, and Erdős's exact bound n^2/2 fails for every n.
Asks whether a monic polynomial with roots in the unit disc has modulus below one on a set of area at least an inverse power of n; proved by Pommerenke (1961), with a constant over log n by Krishnapur, Lundberg and Ramachandran.
Estimates how many Abelian subgroups are needed to cover a group in which every set of more than n elements contains two distinct commuting elements.
Asks whether an order type whose two-colorings always give a red copy of itself or a blue triangle must likewise force a blue complete graph on n vertices.
Asks whether the maximum modulus on the unit circle of the partial products of z minus a unimodular point is unbounded, exceeds a power of n, and has partial sums above n^(1+c); yes by Wagner, Beck, and Korsky with GPT 5.6-Pro.
Asks whether, for every infinite set of reals, some set of positive measure contains no affine copy of it.
Asks whether the largest subset of the first N integers with no five (or no fixed odd number, at least five, of) distinct elements multiplying to a square has size nearly N; Tao (2024) disproved it for every size at least 4.
Characterizes which number theoretic functions f make the shifted values n plus f of n cluster unboundedly in short intervals infinitely often.
Asks whether, for pairwise coprime a, b, c above one, every large integer is a sum of distinct products of powers of a, b and c, none dividing another. Erdős also suggested the weaker hypothesis that a, b and c have no common factor, under which the answer is no, as 6, 10 and 15 show.
Asks whether all large integers are sums of one number from each of several sets of sums of distinct powers of given bases satisfying a density condition.
Asks whether the sumset of integers using only digits zero and one in base three and those using only those digits in base four has positive lower density.