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
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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.
Triangles of area one in sets of large measure
Show that every planar set of large enough measure contains the vertices of a triangle of area 1; Erdős guessed area 4π/√27 suffices.
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 sunflower conjecture
Prove that every family of more than c^n sets of size n contains three sets that pairwise meet in the same core.
The sum-product problem for integers
Show that every finite set of integers has nearly |A|² distinct sums or nearly |A|² distinct products; false over the reals since 2026.
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.
The Littlewood conjecture
Prove that n‖nα‖‖nβ‖ gets arbitrarily close to 0 for every pair of real numbers α and β.
Covering a polynomial lemniscate by discs
Show that the set where a monic polynomial has modulus at most 1 can be covered by discs whose radii sum to at most 2.
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)}.
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–Turán conjecture on additive bases
Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.
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.
The Erdős similarity problem
Show that for every infinite set A of reals some set of positive measure contains no scaled and shifted copy of A.
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.
The Erdős–Hajnal conjecture
Show that graphs avoiding a fixed induced subgraph H always contain a clique or independent set of size n^{c(H)}.
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.
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.