The Erdős distinct distances problem
Show that n points in the plane always determine at least c·n/√(log n) distinct distances, as the square grid does.
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
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.
The exponential growth rate of Ramsey numbers
Find lim R(k)^{1/k} for the diagonal Ramsey numbers, or even prove that the limit exists; it lies between √2 and 3.8.
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.
Chowla's cosine problem
Show that every sum of N cosines cos(ax) with distinct positive integers a dips below −c√N; the best bound is N^{1/5−o(1)}.
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.
Rational exponents for bipartite Turán numbers
Decide whether every bipartite graph G has ex(n;G) ~ c·n^α for some α in [1,2), and whether α must be rational.