Unit circles through three of n points
Show that n points in the plane have only o(n²) unit circles passing through three of them; the best construction gives n^{3/2}.
问题
开放问题和研究问题，每个都是一个 git 仓库：很多来自精选合集，并注明出处。路线是任何人都能开启或接续的分支；检验通过后，结果才算数。
符合的问题
Turán's tetrahedron problem
Determine the largest fraction of triples on n points with no 4-set covered by all its triples; Turán conjectured 5/9.
Making a triangle-free graph bipartite
Show that any triangle-free graph on 5n vertices becomes bipartite after deleting n² edges, or find one that needs more.
Gyárfás's tree packing conjecture
Show that any trees T_2, …, T_n with T_k on k vertices can be packed edge-disjointly into the complete graph K_n.
Doubly exponential growth of 3-uniform Ramsey numbers
Show that the 2-colour Ramsey number of triples, R_3(n), grows doubly exponentially in n; the best lower bound is only 2^{cn²}.
The sparse ruler constant
Find the limit of F(N)/√N, where F(N) is the fewest marks on a ruler of length N that measures every distance; it lies in [1.56, √3].
Singmaster's conjecture on repeated binomial coefficients
Decide whether, for every t, some number appears as a binomial coefficient C(n,k) with k ≤ n/2 in exactly t ways.
The largest Sidon set in {1, …, N}
Decide whether the largest Sidon set in {1,…,N} has √N + O(N^ε) elements; the known error term is of order N^{1/4}.
Representation functions growing exactly like log n
Is there a set A of naturals whose number of representations n = a + b is asymptotic to c·log n for some c > 0?
An asymptotic formula for the Ramsey number R(3,k)
Find the constant c in R(3,k) ~ c·k²/log k; it is now known to lie between 1/2 and 1.
Odd weird numbers
Find an odd number whose proper divisors sum to more than it but no subset sums to it exactly, or prove none exists.
Covering systems with odd moduli
Find a finite set of congruences with distinct odd moduli that covers every integer, or prove that none exists.
The Erdős minimum overlap problem
Pin down the constant c ≈ 0.38 such that every split of {1,…,2N} into equal halves has a difference occurring cN times.
Integer distances in general position
Find 8 or more points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.
The Erdős–Szekeres happy ending problem
Prove that any 2^{n−2}+1 points in general position in the plane contain a convex n-gon; open from n = 7 on.
Lines through four points when no five are collinear
Show that n points with no five on a line determine only o(n²) lines through four of them; constructions reach n^{2−o(1)}.
Explicit Ramsey graphs
Construct explicit graphs on n vertices with no clique or independent set of size c·log n, matching Erdős's random graphs.
The Erdős–Ulam problem on rational distances
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
The Erdős–Straus conjecture
Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.
Sparse halves in triangle-free graphs
Show that every triangle-free graph on n vertices has ⌊n/2⌋ vertices spanning at most n²/50 edges, or find a counterexample.
The Erdős matching conjecture
Determine the largest r-uniform hypergraph on n vertices with no k disjoint edges; Erdős conjectured one of two simple families wins.
Cycles of length a power of two (the Erdős–Gyárfás conjecture)
Show that every finite graph with minimum degree 3 has a cycle of length 4, 8, 16 or another power of two, or find a counterexample.
Integers not overtaken by m + τ(m)
Find n > 24 such that m + τ(m) ≤ n + 2 for every m < n, where τ counts divisors, or prove that none exists.
Dense infinite Sidon sets
Construct an infinite Sidon set with at least N^{1/2−ε} elements up to N for every ε > 0, or show that none exists.
Distinct distances from a vertex of a convex polygon
Show that some vertex of every convex n-gon has at least ⌊n/2⌋ distinct distances to the other vertices, or find a counterexample.
The chromatic number of the plane
Decide whether 6 or 7 colours are needed to colour the plane so that points at distance 1 always get different colours.
Four-cycle-free subgraphs of the hypercube
Show that a subgraph of the n-cube with no 4-cycle has at most about half of the cube's edges; the best bound is 0.60318.
The Brocard–Ramanujan equation n! + 1 = m²
Prove that n! + 1 is a perfect square only for n = 4, 5 and 7, or find a fourth solution.