Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
19 problems
Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and…
What is the maximum volume of a convex body in R^n whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.
How dense can a sphere packing in R^n be as n → ∞? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.
Lassak conjectured that a reduced planar convex body of thickness Δ has area at most (π/4)Δ^2, the value for the disc. False: an explicit reduced body of thickness 1 has area 0.786215… > π/4 = 0.785398…, given by a closed-form support…
Conjectured upper bound on how many pairs among n points in the plane can be exactly one unit apart.
For an arrangement of n hyperplanes in P^3_C with ℓ intersection lines and p intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects p - ℓ + n + 2 ≥ 0. An explicit arrangement built from…
If f(n) is the maximum total side length of n interior-disjoint squares packed in the unit square, is f(k^2 + 1) = k? An exact rational configuration packs 17 squares with total side length greater than 4, refuting the identity at k = 4.
Wegner conjectured in 1965 that every finite family R of axis-parallel rectangles satisfies τ(R) ≤ 2ν(R) - 1, where τ is the minimum number of piercing points and ν the largest pairwise-disjoint subfamily. False, by an explicit…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For a family F of an odd number n of unit disks in the plane, let OA(F) be the area covered by an odd number of disks. It was conjectured that OA(F) ≥ π, the area of a single disk. False: configurations exist with smaller odd area.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For odd k with gcd(n,k) = gcd(n+1,k) = 1, is N_k(n) ≡ ⌊ (k+1)/4 ⌋ pmod 2, where N_k(n) counts pairs 1 ≤ b_i ≤ (k-1)/2 with b_1 + b_2 ≥ (k+1)/2 and b_2 ≡ n b_1 pmod k? Conjectured by Chen and Gendron; its proof removes a conditional step in…
For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be - in particular, can the chromatic number be infinite? Yes: there is such a set, no three…
For the least cutoff c(n) after which every k occurs as the number of homothetic cubes in a decomposition of the unit n-cube, is c(n) ≫ n^n? The Lean proof shows c(n) = o(n^n) along odd dimensions.
If n planar points have no four concyclic, must some point determine (1 - o(1))n distinct distances? Failing that, can one always force more than (1/3 + c)n?
How large can the difference between the largest and second-largest distance multiplicities be among n planar points?
The lattice triangle problem asks which rational triangles unfold to Veech surfaces; in the hard obtuse window it is conjectured that none do. Via an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, the paper rules out…